Stéphane R Louboutin
email: stephane.louboutin@univ-amu.fr

Marc Munsch
email: munsch@math.tugraz.at

Second moment of Dirichlet L-functions, character sums over subgroups, and upper bounds on relative class numbers

We prove an asymptotic formula for the mean-square average of L-functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums A(p, d) recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p -1. We obtain an asymptotic formula for A(p, d) which holds true for any odd divisor d of p -1, thus removing E. Elma's restrictions on the size of d. This answers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range 1 ≤ d ≤ log p 3 log log p we obtain the significant improvement h - p,d ≤ 2 (1+o(1))p 24 m/4

over the trivial bound h - p,d ( dp 24 ) m/4 on the relative class numbers of the imaginary number fields of conductor p ≡ 1 mod 2d and degree m = (p -1)/d, where d ≥ 1 is odd.

Introduction

Throughout the paper d ≥ 1 will be an odd integer and p will be an odd prime satisfying p ≡ 1 (mod 2d). We also write log j for the j-th iterated logarithm. We let X p denote the multiplicative cyclic group of order p -1 of the Dirichlet characters mod p and let X * p denote the set with p -2 elements of the non-trivial Dirichlet characters mod p. Whenever m divides p -1, we let χ p,m denote any one of the φ(m) characters in X p of order m. Notice that χ p,m is odd, i.e. such that χ p,m (-1) = -1, if and only if d = (p -1)/m is odd, which implies that m is even. Let h - p,d be the relative class number of the imaginary subfield K p,d of the cyclotomic field Q(ζ p ) of even degree (K p,d : Q) = m and odd relative degree (Q(ζ p ) : K p,d ) = d (e.g. see [Was,Chapter 4]). For d = 1, we have K p,1 = Q(ζ p ) and it has long been known that

h - p,1 = h - Q(ζp) ≤ 2p p 24 (p-1)/4 = 2p p 24 m/4 , (1)
see [Met] and [Wal]. See also [START_REF] Louboutin | Quelques formules exactes pour des moyennes de fonctions L de Dirichlet[END_REF] where it is explained how to get explicit bounds better than (1) of the type h - p,1 ≤ 2p(1 + o(1))(p/C) (p-1)/4 for any constant C greater than 4π 2 . Notice also that by [AC] we have h - p,1 = 2p 1+o(1) p 4π 2

(p-1)/4 and see [Gra] for more subtle results according to which Kummer's conjecture on the asymptotic behavior h - p,1 ∼ 2p p 4π 2

(p-1)/4 is unlikely to be true. Denote by w p,d the number of complex roots of unity contained in K p,d ; we have w p,1 = 2p and w p,d = 2 for d > 1. The following bound holds (e.g. see [Was,Chapter 4,page 42] for the equality and then use the arithmetic-geometric mean inequality):

h - p,d = w p,d m/2 j=1 √ p 2π L(1, χ 2j-1 p,m ) ≤ w p,d pM (p, m) 4π 2 m/4 , (2)
where M (p, m) denotes the following mean square of L(1, χ) as χ runs over the m/2 odd characters in the only subgroup of order m of X p :

M (p, m) := 2 m m/2 j=1 |L(1, χ 2j-1 p,m )| 2 . (3)
Therefore explicit formulas (or asymptotic formulas) for M (p, m) allow to give precise upper bounds of type h - p,d ≤ C 1 • C m/4 2

. For d = 1, H. Walum deduced (1) in [Wal] by proving that

M (p, p -1) = π 2 6 1 - 1 p 1 - 2 p . (4)
For d = 3, 5, some explicit formulas for M (p, m) have been obtained in certain cases by the first author (see Section 2.2) allowing him to give upper bounds on h - p,3 and h - p,5 . These results will be generalized in Proposition 7 below. In all these situations M (p, (p -1)/d) is asymptotic to π 2 /6. It is unrealistic to obtain an explicit formula for all values of d and the following simple argument gives a trivial bound on M (p, m). Since in M (p, m) we consider only m/2 of the (p -1)/2 odd Dirichlet characters that appear in M (p, p -1), we have

M (p, m) ≤ 2 m p-1 2 M (p, p -1) = dM (p, p -1) = d π 2 6 1 -1 p 1 -2 p . (5)
By (2), this implies that

h - p,d ≤ 2 dp 24 p-1 4d = 2 dp 24 m/4 . ( 6
)
For a given m it is hopeless to get an asymptotic on M (p, m). Indeed, it is reasonable to conjecture that for a given even integer m there are infinitely many primes p ≡ 1 + m (mod 2m) for which M (p, m) (log log p) 2 and infinitely many primes p ≡ 1 + m (mod 2m) for which M (p, m) (log log p) -2 (see [Lou16, bottom of page 166 and top of page 167]).

The aim of the paper is to give an asymptotic formula for M (p, m) when m is of reasonable size with respect to p and an upper bound when m is small. As a consequence we obtain a significant improvement upon the trivial bound (6).

Theorem 1 As p tends to infinity and d ≥ 1 runs over the odd divisors of p -1 such that 1 ≤ d ≤ log p 3 log log p , we have the asymptotic formula

M (p, m) = M (p, (p -1)/d) = π 2 6 1 + O(d(log p) 2 p -1 d-1 ) = π 2 6 (1 + o(1)) , (7)
which implies the upper bound

h - p,d ≤ 2 (1 + o(1))p 24 m/4 . ( 8
)
If the previous bound does not apply but d is such that log d = o(log p/ log 2 p), we have for some absolute constant C > 0

M (p, m) = M (p, (p -1)/d) ≤ C(log 2 d) 2 (9)
which implies, in this range of d, the upper bound

h - p,d ≤ (Cp(log 2 d) 2 ) m/4 . ( 10
)
Remarks 2 The asymptotic formula (7) answers a question raised in [Lou16, Section 6, Question 2]. It should be emphasized that the error term in (7) is almost optimal, in view of Proposition 8. Furthermore (8) is in accordance with the known asymptotics (see [START_REF] Louboutin | A finitness theorem for imaginary abelian number fields[END_REF]Theorem 4])

log h - p,d ∼ m+o(1)
4 log p. For very large d, the bound M (p, m) log 2 p remains the best known. This is not surprising if we look at the very extreme case d = (p -1)/2 and p ≡ 3 (mod 4). In that situation, χ is the quadratic character given by the Legendre symbol χ(n) = n p and [Lit] proved that L(1, χ) log 2 p but improving upon the bound L(1, χ) log p remains out of reach unconditionally. On the other hand we cannot expect a uniform bound for M (p, m) better than (log 2 p) 2 . Indeed, Chowla [Cho] proved unconditionally that there are infinitely many quadratic characters χ such that L(1, χ) log 2 p. This supports the hypothesis that the bound (9) could be sharp.

M (p, m) = |L(1, χ)| 2 . Under GRH, Littlewood
Finally, let d be an odd integer and p ≡ 1 (mod 2d) be a prime integer. Let χ be an odd Dirichlet character of (even) order m = (p -1)/d dividing p -1 and prime conductor p ≥ 3. Set

A(p, d) = 1 p -1 p-1 N =1    1≤n 1 ,n 2 ≤N χ(n 1 )=χ(n 2 ) 1    (11)
(the results depend only on p and d, not on the choice of χ). Then the paper is organized as follows. To begin, in Section 2, we recall (and give a simple proof of) a formula discovered by Elma relating M (p, m) to the character sums A(p, d) defined in (11). In Section 3, Proposition 8, we show that for a certain family of primes p and d we can compute exactly M (p, m) using properties of Dedekind sums. Finally, in Section 4, we answer a question of Elma and prove the following asymptotic formula for A(p, d) which directly implies Theorem 1 (see Section 5):

Theorem 3 As p tends to infinity and d ≥ 1 runs over the odd divisors of p -1, we have

A(p, d) = (2d + 1)p 6 + o(dp).
A crucial point of the analysis comes from the fact that the average in (3) is made over a family of m/2 characters which could be of any size with respect to p. The same difficulty carries into the analysis of A(p, d) with a character sum averaged over a subgroup of F * p of size d. On the one hand when d is small (see Theorem 10) we write A(p, d) as an average of a function evaluated at equidistributed points modulo 1 and use techniques from discrepancy theory. On the other hand, when d is large (see Theorem 19) we rely on character sums techniques and incorporate recent estimates on the frequency of large character sums [BGGK].

2 Elma's character sums 2.1 Elma's character sums and the mean square value M (p, m) E. Elma proved a nice connection between the mean square values M (p, m)'s and these character sums A(p, d). We give a simple and short proof of [START_REF] Elma | On a problem related to discrete mean values of Dirichlet L-functions[END_REF]Theorem 1.1]:

Theorem 4 Let χ be a primitive Dirichlet character modulo f > 2, its conductor. Set S(k, χ) = k l=0 χ(l) and let L(s, χ) = n≥1 χ(n)n -s be its associated Dirichlet L-series. Then

f -1 k=1 |S(k, χ)| 2 = f 2 12 p|f 1 - 1 p 2 + a χ f 2 π 2 |L(1, χ)| 2 , where a χ := 0 if χ(-1) = +1, 1 if χ(-1) = -1.
Proof. Our simple proof is based on an easy to remember idea: we apply Parseval's formula

1 0 |F (x)| 2 dx = ∞ n=-∞ |c n (F )| 2 to the function x ∈ [0, 1) → F (x) := 0≤l≤f
x χ(l) extended to x ∈ R by 1-periodicity. The reader would be able to reconstruct the argument using this simple idea. Let us now give all the details. Since χ is primitive, the Gauss sums τ (n, χ)

= f k=1 χ(k) exp(2πink/f ) and τ (χ) = τ (1, χ) satisfy τ (n, χ) = χ(n)τ (χ) and |τ (χ)| 2 = f , e.g.
see [Was,Lemmas 4.7 and 4.8]. (These properties are easy to check when f

= p ≥ 3 is prime). Since x → F (x) = S(k, χ) is constant for x ∈ [k/f, (k + 1)/f ), we have 1 0 |F (x)| 2 dx = f -1 k=0 (k+1)/f k/f |F (x)| 2 dx = f -1 k=0 (k+1)/f k/f |S(k, χ)| 2 dx = 1 f f -1 k=0 |S(k, χ)| 2 ,
and the n-th Fourier coefficient of F is given by

c n (F ) = 1 0 F (x)e -2πinx dx = f -1 k=0 (k+1)/f k/f F (x)e -2πinx dx = f -1 k=0 S(k, χ) (k+1)/f k/f e -2πinx dx.
Hence, by [Was,Theorem 4.2] we have

c 0 (f ) = 1 f f -1 k=0 S(k, χ) = 1 f f -1 k=0 k l=0 χ(l) = 1 f f -1 l=0 (f -l)χ(l) = - 1 f f -1 l=0 lχ(l) = L(0, χ)
and for n = 0 we have [Was,Chapter 4,page 30]. Therefore, Parseval's formula gives

c n (F ) = f -1 k=0 S(k, χ) exp -2πin(k+1) f -exp -2πink f -2πin = f -1 k=1 (S(k, χ) -S(k -1, χ)) exp -2πink f 2πin (notice that S(0, χ) = S(f -1, χ) = 0) = τ (-n, χ) 2πin = τ (χ) 2πi × χ(-n) n = -χ(-1)c -n (F ). Now, L(0, χ) = 0 if χ(-1) = +1 and |L(0, χ)| 2 = f π 2 |L(1, χ)| 2 if χ(-1) = -1, e.g. see
1 f f -1 k=0 |S(k, χ)| 2 = a χ f π 2 |L(1, χ)| 2 + 2 n≥1 gcd(n,f )=1 f 4π 2 × 1 n 2 = a χ f π 2 |L(1, χ)| 2 + f 12 p|f 1 - 1 p 2
and the desired result follows. Notice that this proof is similar to the ones in [BC].

•
Corollary 5 Let χ be an odd Dirichlet character of (even) order m dividing p -1 and prime conductor p ≥ 3. Set d = (p -1)/m (an odd integer) and let A(p, d) be as in ( 11). Then

M (p, m) := 2 m m-1 j=0 j odd |L(1, χ j )| 2 = π 2 6 p -1 p 2 (12A(p, d) -(4d + 1)p -d -1) . ( 12
)
In particular,

(4d + 1)p + d + 1 12 ≤ A(p, d) ≤ (5d + 1)p + d + 1 12 . ( 13
)
Moreover, by [Lou96a], we have

0 ≤ M (p, m) ≤ (log p + 2 + γ -log π) 2 /4. ( 14
)
Proof. By Theorem 4, for j odd we have

|L(1, χ j )| 2 = - π 2 12 1 - 1 p 2 + π 2 p 2 p-1 k=1 |S(k, χ j )| 2
The χ j 's are primitive modulo p for 1 ≤ j ≤ m -1, whereas χ 0 is the non-primitive trivial Dirichlet character modulo p. Therefore, on the one hand we have

m-1 j=0 p-1 k=1 |S(k, χ j )| 2 = m-1 j=0 p-1 k=1 k l=1 χ j (l) 2 = m-1 j=0 p-1 k=1 1≤l 1 ,l 2 ≤k χ j (l 1 )χ j (l 2 ) = m(p -1)A(p, d),
by using the orthogonality relation

m-1 j=0 χ j (n 1 )χ j (n 2 ) = m if χ(n 1 ) = χ(n 2 ) = 0, 0 otherwise.
On the other hand, Theorem 4 gives

m-1 j=1 p-1 k=1 |S(k, χ j )| 2 = (m -1)(p 2 -1) 12 + mp 2 2π 2 M (p, m). Since p-1 k=1 |S(k, χ 0 )| 2 = p-1 k=1 k l=1 1 2 = p-1 k=1 k 2 = (p -1)p(2p -1) 6 , it follows that m(p -1)A(p, d) = m(p 2 -1) 12 + (p -1) 2 (4p + 1) 12 + mp 2 2π 2 M (p, m).
The desired identity (12) follows. Now, noticing that M (p, m) ≥ 0, the lower bound on A(p, d) in ( 13) follows from ( 12). Finally, by ( 5) we have

M (p, m) ≤ dπ 2 6 1 - 1 p .
Plugging this bound in ( 12) we obtain the upper bound on A(p, d) in ( 13). 18) below. Hence it should be possible to improve upon the upper bound in (13).

Exact formulas for M (p, m) and A(p, d) in specific cases

There are only four cases listed below where an explicit formula for A(p, d) is known.

1. By ( 4) and ( 12), for d = 1 we have

A(p, 1) = p 2 = (2d + 1)p 6 . ( 15
) 2. For d = 3 we proved in [Lou16, Theorem 1] that M (p, (p -1)/3) = π 2 6 1 - 1 p (for p ≡ 1 (mod 6)) (16)
and the corresponding bound on the relative class number

h - p,3 ≤ 2 p 24 (p-1)/12 = 2 p 24 m/4 . ( 17
)
By ( 12), this gives for d = 3 and p ≡ 1 (mod 6),

A(p, 3) = 7p + 2 6 = (2d + 1)p 6 + o(p). (18)
3. For d = 5 we proved in [START_REF] Louboutin | Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers[END_REF]Theorem 5] that M (p, (p -1)/5) = π 2 6 1 + 2a(a + 1) 2 -1 p (for p > 5 of the form p = a 5 -1 a-1 ). ( 19)

and the corresponding bound on the relative class number

h - p,5 ≤ 2 p 24 (p-1)/20 = 2 p 24 m/4 . ( 20
)
By ( 19) and ( 12), this implies

A(p, 5) = 11p + 3 6 + a(a + 1) 2 p 6(p -1) = (2d + 1)p 6 + o(p). (21)
4. For d = (p -1)/2 and 3 < p ≡ 3 (mod 4). In that situation, χ is the quadratic character given by the Legendre symbol

χ(n) = n p , L(1, χ) = πh Q( √ -p) / √ p and (12) gives A(p, (p -1)/2) = 4p 2 -p + 1 24 + p 2 2π 2 (p -1) |L(1, χ)| 2 = 4p 2 -p + 1 24 + ph 2 Q( √ -p) 2(p -1) = dp 3 + O(p log 2 p),
where we used the bound |L(1, χ)| log p.

Remarks 7 In fact, d = 1 is the only case for which we could come up with a direct proof of the formula for A(p, d). Indeed, we have χ p,p-1 (n 1 ) = χ p,p-1 (n 2 ) if and only if n 1 ≡ n 2 (mod p). Hence,

A(p, 1) = 1 p -1 p-1 N =1 N = p 2 .
It would be nice to have similar independent and direct proofs of ( 18) and ( 21).

3 Evaluation of M (p, m) for primes p = (a d -1)/(a-1) ≡ 1 (mod 2d)

We gave an explicit formula for A(p, 3), see ( 18), and one for A(p, 5), but only for the primes p of the form p = (a 5 -1)/(a -1), see (21). After some numerical computation for primes of the form (a 5 -2 5 )/(a -2) or (a 5 -3 5 )/(a -3) we could not guess any formula for M (p, (p -1)/5) or A(p, 5). However, we now prove a general result which recover ( 16) and ( 19) (let us say that we forgot to deal with the case a < 0 in the proof of [START_REF] Louboutin | Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers[END_REF]Theorem 5]). We want to point out that here again we do not directly compute A(p, d). Instead we give an exact formula for M (p, m) and then use (12) to deduce an expression for A(p, d).

Proposition 8 Set Q l (X) = (X l -1 -l(X -1))/(X -1) 2 ∈ Z[X], l ≥ 1. Hence, Q 1 (X) = 0, Q 2 (X) = 1 and Q l (X) = X l-2 + 2X l-3 + • • • + (l -2)X + (l -1) for l ≥ 2. Let d ≥ 3 be a prime integer.
For a prime integer of the form p = (a d -1)/(a -1) for some a = -1, 0, 1, we have

M (p, (p -1)/d) = π 2 6 1 + 2a(a + 1) 2 Q d-1 2 (a 2 ) -1 p , (22)
A(p, d) = (2d + 1)p + d+1 2 6 + p p -1 • a(a + 1) 2 Q d-1 2 (a 2 ) 6 = 2d + 1 6 p + O p 1-1 d-1 , ( 23
)
and h - p,d ≤ 2 (p/24) m/4 for p = (a d -1)/(a -1) with a ≤ -2.
Proof. We keep the notation of [START_REF] Louboutin | Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers[END_REF], use the properties of Dedekind sums

s(c, d) = 1 4d |d|-1 n=1 cot πn d cot πnc d (c ∈ Z, d ∈ Z \ {-1, 0, 1})
recalled in [START_REF] Louboutin | Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers[END_REF] and set l = (d -1)/2. To deal in one stroke with the two cases a ≤ -2 and a ≥ 2 we have extended the definition of Dedekind sums, allowing d to be negative. The reciprocity and complementary laws for these generalized Dedekind sums are

s(c, d) + s(d, c) = c 2 + d 2 -3|cd| + 1 12cd and s(1, d) = d 2 -3|d| + 2 12d .
Set p k = (a k -1)/(a -1). By [Lou16, Corollary 3] we have

M (p, (p -1)/d) = π 2 6 1 + N p , where N = 24 l k=1 s(a k , p) -3 + 2 p . Now, p ≡ p k (mod a k ) and a k ≡ 1 (mod p k ) for 1 ≤ k ≤ d. Hence s(a k , p) = a 2k + p 2 -3|a| k p + 1 12a k p -s(p, a k ) = a 2k + p 2 -3|a| k p + 1 12a k p -s(p k , a k ) and s(p k , a k ) = p 2 k + a 2k -3|p k a k | + 1 12p k a k -s(a k , p k ) = p 2 k + a 2k -3|p k a k | + 1 12p k a k -s(1, p k ),
by the reciprocity law for Dedekind sums. Since

s(1, p k ) = p 2 k -3|p k | + 2 12p k ,
by the complementary law for Dedekind sums, we obtain

s(a k , p) = a 2k + p 2 -3|a| k p + 1 12a k p - p 2 k + a 2k -3|p k a k | + 1 12p k a k - p 2 k -3|p k | + 2 12p k .
The contribution of the three terms with absolute values is equal to

- k 4 + 2k+1 4 - k+1 4
, where = a/|a| is the sign of a and therefore k+1 is the sign of p k . Hence, this contribution is equal to -1 4 whatever the value of ∈ {-1, 1}. Since this contribution does not depend on , we may get rid of the absolute values and we obtain

s(a k , p) = a 2k + p 2 -3a k p + 1 12a k p - p 2 k + a 2k -3p k a k + 1 12p k a k - p 2 k -3p k + 2 12p k = a 2k + p 2 -3a k p + 1 12a k p - p k -3a k 12a k - p k -3 12 + a 2k + 1 12a k p k - 2 12p k = a 2k + p 2 -3a k p + 1 12a k p - p k (1 -a k ) 12a k + (a k -1) 2 12a k p k = a 2k + p 2 -3a k p + 1 12a k p - (1 -a)p 2 k 12a k + (a -1)(a k -1) 12a k and s(a k , p) = a 2k + p 2 -3a k p + 1 12a k p + (a -1) p 2 k + 1 -a k 12a k .
Notice that the more natural congruence p ≡ p k-1 (mod a k ) and a k ≡ a (mod p k-1 ) would lead to slightly more complicated computations. An easy but boring computation using l k=1 b k = b(b l -1)/(b -1) then finally yields N = 2a(a + 1) 2 Q l (a 2 ) -1, as desired. • Remarks 9 It is a long standing conjecture that there are infinitely many primes of the form p = (a d -1)/(a -1), as first investigated in the special case of Mersenne primes 2 p -1 (a = 2). More precise results about the number of such primes less than x are expected. This is sometimes called the Lenstra-Pomerance-Wagstaff conjecture (see the survey [Pom] for more information and references on this topic).

Asymptotic behavior of Elma's sums and proof of Theorem 3

Let us remark that

A(p, d) dp/3 = pM (p, m) 2π 2 d(p -1) + 1 + 1 4d + 1 4p + 1 4dp
, by (12). Hence, as d/ log 2 p tends to infinity (or equivalently m = o(p/ log 2 p)), we have by ( 14)

A(p, d) ∼ dp 3 , (24)
as noticed in [Elm]. However, according to the results of Section 2.2, for a given d we cannot expect this asymptotic but rather the refined asymptotic

A(p, d) ∼ (2d + 1)p 6 . ( 25
)
Our goal in this section is to prove Theorem 3, i.e. that (25) indeed holds true without any restriction on the size of the parameter d.

We will split the discussion into two cases depending on whether d goes or not to infinity. Theorem 3 follows from Theorems 10 (d small) and 19 (d large) proved below. In the former case, we obtain the more precise asymptotic expansion A(p, d) ∼ (2d+1)p 6 . By ( 12), this allows us to deduce an asymptotic formula for M (p, m). In the latter case, Theorem 3 is not sufficient to infer an asymptotic formula for M (p, m) and only implies an upper bound.

Asymptotic for small d's

Our goal in this section is to prove the following theorem which gives Theorem 3 for small d's:

Theorem 10 Let d range over the odd integers. Set γ(d) = max k|d φ(k) ≤ d -1. Let p range over the prime integers such that p ≡ 1 (mod 2d). Then we have the following asymptotic formula

A(p, d) = 2d + 1 6 p + O d(log p) 2 p 1-1/γ(d) = 2d + 1 6 p + O dp(log p) 2 p -1/(d-1)
where the implicit constant in the error term is absolute.

In particular, in the range 1 ≤ d ≤ log p 3 log log p , we have

A(p, d) ∼ 2d + 1 6 p.
Remarks 11 Observe that γ(d) = d -1 whenever d is an odd prime. Hence by ( 23) the power of p in the error term of Theorem 10 is optimal.

Results from uniform distribution theory

For any fixed integer s, we consider the s-dimensional cube I s = [0, 1] s equipped with its sdimensional Lebesgue measure λ s . We denote by B the set of rectangular boxes of the form

s i=1 [α i , β i ) = {x ∈ I s , α i ≤ x i < β i } where 0 ≤ α i < β i ≤ 1.
If S is a finite subset of I s , we define the discrepancy D(S) by

D(S) = sup B∈B #(B ∩ S) #S -λ s (B) .
The discrepancy measures in a quantitative way the deviation of a pointset S from equidistribution. In particular a sequence of sets S n is uniformly distributed if and only if D(S n ) ---→ n→∞ 0. In order to state a quantitative version of this phenomenon known as the Koksma-Hlawka inequality, the concept of functions of bounded Hardy-Krause variation is used. In words, the Hardy-Krause variation is the sum of the Vitali variations1 of all the restrictions of f to the faces of I s .

Theorem 12 [START_REF] Drmota | Sequences, discrepancies and applications[END_REF]Theorem 1.14] Let f (x) a function of bounded variation on I s in the sense of Hardy and Krause and x 1 , . . . , x N a finite sequence of points in I s . Then

1 N N i=1 f (x i ) - Is f (u)dλ s (u) ≤ V (f )D(S)
where V (f ) is the Hardy-Krause variation of f (see also [START_REF] Kuipers | Uniform distribution of sequences[END_REF]Chapter 2]).

In order to estimate the discrepancy, we recall the inequality of Erdős-Turán-Koksma:

Theorem 13 [START_REF] Drmota | Sequences, discrepancies and applications[END_REF]Theorem 1.21]. Let S = {x 1 , . . . , x N } be a set of points in I s and H a positive integer. Then we have

D(S) ≤ 3 2 s   2 H + 1 0< h ∞ ≤H 1 r(h) 1 N N n=1 e( h, x N )   , (26)
where e(z) = exp(2πiz), r(h) = s i=1 max{1, |h i |} for h = 1 , . . . , h s ) ∈ Z s and , denotes the standard inner product in R s .

Notions from pseudo random generators theory

In the rest of the paper the results of the previous section will only be used for s = 2. We introduce some tools from the theory of pseudo-random generators and optimal coefficients in a very basic situation. We refer for more information to the survey of Korobov [Kor], the work of Niederreiter [START_REF] Niederreiter | Pseudo-random numbers and optimal coefficients[END_REF][START_REF] Niederreiter | Quasi-Monte Carlo methods and pseudo-random numbers[END_REF] or the book of Konyagin and Shparlinski [START_REF] Konyagin | Character sums with exponential functions and their applications[END_REF]Chapter 12] and keep their notations. For any prime p and integer 1 ≤ λ ≤ p -1 we define

σ(λ, p) := 0< h ∞ ≤p-1 δ p (h 1 + h 2 λ) r(h)
where δ p (a) = 1 if a = 0 mod p and δ p (a) = 0 otherwise. For any λ, we define ρ(λ, p) = min

h =0 r(h)
where the min is taken over all non trivial solutions h = (h 1 , h 2 ) of the congruence

h 1 + h 2 λ = 0 mod p.
The next lemma shows that 1 ρ(λ,p) and σ(λ, p) are relatively close to each other:

Lemma 14 [Nied77, Theorem 3.8]. There exists C > 0 such that, for any prime p ≥ 3, and λ ∈ {1, . . . , p -1} we have

1 ρ(λ, p) ≤ σ(λ, p) ≤ C (log p) 2 ρ(λ, p) . (27)
In some cases which are of interest for our problem, we can control from below ρ(λ, p):

Lemma 15 Let λ be an element order k ≥ 3 in the multiplicative group F * p . Then

ρ(λ, p) ≥ p 1/φ(k) / √ 8,
where φ denotes as usual the Euler's totient function.

Proof. Let Φ k (X) = 0≤l≤φ(k) a l X l = 1≤l≤k gcd(k,l)=1 (X -ζ l k ) denote the k-th cyclotomic polynomial. Set Φ k (X) 2 =   0≤l≤φ(k) |a l | 2   1/2 = 1 2π 2π |Φ k (e it )| 2 dt 1/2 ≤ 2 φ(k) .
We clearly have Φ k (λ) = 0 mod p. For h = (h 1 , h 2 ) = 0 we define P (X) = h 1 + h 2 X. Assume that P (λ) = 0 mod p, then p divides the resultant R = Res(P, Φ k ). The polynomial Φ k being irreducible of degree ≥ 2, we deduce that R = 0. It follows that |R| ≥ p. Since R is the determinant of the Sylvester matrix of P (X) and Φ k (X), by Hadamard's inequality we have

|R| ≤ P (X) deg Φ k (X) 2 Φ k (X) deg P (X) 2 ≤ h 2 1 + h 2 2 φ(k)/2 2 φ(k) ≤ (max(|h 1 |, |h 2 |)) φ(k) 8 φ(k)/2 . Hence we have r(h) ≥ max(|h 1 |, |h 2 |) ≥ |R| 1/φ(k) / √ 8 ≥ p 1/φ(k) / √ 8.
All together we obtain the lower bound ρ(λ, p) p 1/φ(k) / √ 8. •

Reduction to a problem of equidistribution

Set H = ker(χ), the subgroup of F * p of order d. We interpret the condition χ(n 1 ) = χ(n 2 ) as n 1 n -1 2 ∈ H. We write H as a disjoint union

H = k|d H k , where H k := {θ ∈ H, ord(θ) = k}.
Proposition 16 For any pair (x, y) of I 2 we define

f d (x, y) = x d -1 + min(x, y).
We have the following relation

A(p, d) = 1 p -1 1≤n 1 ,n 2 ≤p-1 χ(n 1 )=χ(n 2 ) min(n 1 , n 2 ) = p (p -1) k|d k =1 θ∈H k x mod p f d x p , xθ p .
Proof. Changing the order of summation in (11) and making the change of variables (n 1 , n 2 ) → (p -n 1 , p -n 2 ), we do have

(p -1)A(p, d) = 1≤n 1 ,n 2 ≤p-1 χ(n 1 )=χ(n 2 ) (p -max(n 1 , n 2 )) = 1≤n 1 ,n 2 ≤p-1 χ(n 1 )=χ(n 2 ) min(n 1 , n 2 ). Now we have 1≤n 1 ,n 2 ≤p-1 χ(n 1 )=χ(n 2 ) min(n 1 , n 2 ) = x mod p   x + θ∈H θ =1 min(x, θx)    .
We remark that if θ = 1, we have

min(x, θx) = pf d x p , xθ p - x d -1 .
Using the decomposition H = k|d H k and summing over H, the proposition follows. , where λ generates H. This comes from the fact that these points are not equidistributed in I d because they lie in the hyperplane of equation

x 1 + • • • + x d = 0.

Proof of Theorem 10

We introduce the set of points in I 2 :

S θ = x p ,
I 2 f d (u, v)dudv ≤ V (f d )D(S θ ).
It is easy to compute the integral and obtain

I 2 f d (u, v)dudv = 1 2(d -1) + 1 3 .
Applying Proposition 16 and simplifying, we obtain the equation

A(p, d) = 2d + 1 6 p + O (ET ) (28)
where the error term is

ET := pV (f d )    k|d k =1 θ∈H k D(S θ )    . ( 29
)
The readers can easily convince themselves that V (f d ) 1 independently of d. Indeed we have V (f d ) ≤ 1 d-1 V ((x, y) → x) + V (min(x, y)) 1 using basic properties of the Hardy-Krause variation (see for instance [START_REF] Aistleitner | Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality[END_REF]Equation (22)]) and the fact that the function (x, y) → min(x, y) is of bounded Hardy-Krause variation. 2Hence to finish the proof, we need to bound the sum of discrepancies. Applying Theorem 26 with H = p -1 we obtain

D(S θ ) ≤ 3 2 2   2 p + 0< h ∞ ≤p-1 1 r(h) 1 p p x=1 e h 1 x + h 2 xθ p   .
Using the orthogonality relations for an absolute constant C. Hence recalling that γ(d) = max k|d φ(k) and summing over k, we arrive at d) .

ET = pV (f d )    k|d k =1 θ∈H k D(S θ )    d(log p) 2 p 1-1/γ(
This concludes the proof of Theorem 10, in view of Equation (28).

Asymptotic for large d's

For a given non-principal Dirichlet character χ mod p, where p is a prime, let

M (χ) := max 1≤x≤p n≤x χ(n)
and its renormalization

m(χ) = M (χ) e γ √ p/π .
The Pólya-Vinogradov Theorem states that m(χ) log p

for all non-principal characters χ mod p. Apart from some improvements on the implicit constant, this remains the state-of-the-art for the general non-principal character. However, for most of the characters M (χ) is much smaller and we can study how often M (χ) is large. The best result in this direction was obtained in [BGGK]:

Theorem 18 Let η = e -γ log 2. If 1 ≤ τ ≤ log 2 p -M for some M ≥ 4, then Φ p (τ ) := 1 p -1 # {χ mod p : m(χ) > τ } ≤ exp - e τ -2-η τ (1 + O((log τ )/τ )) .
We are now in a position to prove Theorem 3 for large d's. Remarks 20 The condition log d = o(log p/ log 2 p) could be made more explicit by specifying the constants in the proof below. Notice also that whereas Theorem 3 follows from Theorems 10 and 19, it does not follow from Theorem 10 and (24).

Proof. The first part of the Theorem follows directly from (12) and the inequality |L(1, χ) log p. This could also be proved following our argument below and using only the Pólya-Vinogradov inequality. Let us now focus on the case log d = o(log p/ log 2 p). The condition χ(n 1 ) = χ(n 2 ) is equivalent to n 1 n -1 2 lying in the kernel of χ, which is a subgroup of order d of the multiplicative cyclic group F * p . We apply the orthogonality of characters in the subgroup < χ > of order m generated by χ ∈ X p and rewrite the sum A(p, d) defined in (11) as

A(p, d) = 1 (p -1) p-1 N =1 1 m Ψ∈Xp Ψ m =χ 0 1≤n 1 ,n 2 ≤N Ψ(n 1 n -1 2 ).
Separating the contribution of the trivial character from the others, this leads us to the equation Let us set the parameter τ = min{C(log 2 d), log 2 p -M }, where M is the constant appearing in Theorem 18 and C is some large constant which will be specified later. We introduce the following set of characters X τ p,0 = Ψ ∈ X * p : m(Ψ) ≤ τ and further define for every integer 1 ≤ j ≤ J X τ p,j := Ψ ∈ X * p : 2 j-1 τ < m(Ψ) ≤ 2 j τ ,

A(p, d) = d (p -1) 2 p-1 N =1 N 2 + 1 (p -1) p-1 N =1 1 m Ψ∈X * p Ψ m =χ 0 1≤n≤N Ψ(n)
where J is chosen in order to allow an application of Theorem 18. Precisely, we choose J such that τ 2 J ≤ log 2 p -M < τ 2 J+1 .

We now split the characters appearing in the summation in R as follows

X * p =   J j=0 X τ p,j   Ψ ∈ X * p : m(Ψ) > 2 J τ .
Notice that if τ = log 2 p -M then J = 0 and we only split the summation depending on whether m(Ψ) ≤ log 2 p -M or not. Remark that there are at most m characters Ψ ∈ X τ p,0 appearing in the sum. Hence, it follows from Theorem 18 and the inequality (30) that

R p-1 N =1   mpτ 2 + p 2 J j=1
τ 2 2 2j Φ p (τ 2 j-1 ) + p 2 (log p) 2 Φ p (τ 2 J )

  p-1 N =1   mpτ 2 + p 2 J j=1
τ 2 2 2j exp -e τ 2 j-1 100τ 2 j + p 2 (log p) 2 exp -e τ 2 J 100τ 2 J   . (32)

The summation over j in the right hand side of ( 32) is clearly dominated by its first term. Thus we obtain after summing over N and recalling our choice of J: R p 3 d τ 2 + p 3 τ 2 exp -c 1 e τ τ + p 3 (log p) 2 e -c 2 log p/ log log p (33)

for some absolute constants c 1 , c 2 > 0. We insert (33) in (31) and choose C large enough in the definition of τ to ensure that the second and third term in the right hand side of (33) have negligible contribution. This is indeed possible due to the restriction on the size of d and concludes the proof. •

  -1) = dp/3 + O(d).

  Therefore we are left to bound the contribution of non-trivial characters andA(p, d) = dp/3 + O

•

  Remarks 17The reader might wonder why we did not express directly the sum A(p, d) using the more natural function on I d given by g(x 1 , . . . , x d ) = d i=1 min(x 1 , x i ) evaluated at the points

	x p , xλ p , . . . , xλ d-1 p

  Theorem 19 We have A(p, d) = dp/3 + O(p log 2 p), where the implicit constant in this error term is absolute and effective. Moreover, if d and p go to infinity with log d = o(log p/ log 2 p), then we have the better asymptotic

A(p, d) = dp/3 + O(p(log 2 d) 2 ),

where the implicit constant in this error term is absolute and effective. Consequently, if d goes to infinity, then A(p, d) = dp/3 + o(dp).

2020 Mathematics Subject Classification. Primary.

11R42, 11L40. Secondary: 11A07, 11J71, 11M20, 11R18, 11R20. Key words and phrases: moments of Dirichlet L-functions, cyclotomic field, relative class number, character sums, multiplicative subgroups, exponential sums, Dedekind sums, discrepancy.

To have a rough idea in two dimensions, look at the variation of f over the rectangle R := [x1, x2] × [y1, y2], namely vR(f ) := f (x2, y2)-f (x1, y2)-f (x2, y1)+f (x1, y1). The Vitali variation can then be obtained by summing vR(f ) over a partition of I2 and taking the supremum over all possible partitions.

Indeed the Hardy-Krause variation is obtained as a sum of the Vitali variations of min(x, y), min(x, 1) and min(1, y).

Acknowledgements

The second author would like to thank Igor Shparlinski for sketching a refinement of our argument in the proof of Theorem 19 leading to a better result.

Funding

This work was supported (for M. M) by the Austrian Science Fund (FWF) [P-33043].

Proof of Theorem 1

The first part of Theorem 1 follows from Theorem 10 and (12). The second part follows from Theorem 19 and (12).

Concluding remarks

We solved Elma's question about the asymptotic behavior of the character sums A(p, d) regardless of the size of d. As already noticed above, for d large, this is not precise enough to deduce an asymptotic formula for the mean-square value M (p, m). To conclude, let us say that the upper bound ( 9) could be obtained by working directly with L(1, χ) following our method of proof of Theorem 19. This requires results about the distribution of L(1, χ) as the ones obtained by Granville and Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF][START_REF] Granville | Extreme values of ζ(1 + it). The Riemann zeta function and related themes: papers in honor of Professor K. Ramachandra[END_REF] instead of Theorem 18.


Luc

Luc-Yaovi Kouassi

The figure of the entrepreneur: social impact of TEEP beneficiaries (Nigeria, Cameroon and Côte d'Ivoire)

Supporting Entrepreneurship in Africa" Nigeria (1969) Kenya (385) Cameroun (106) Côte d'Ivoire (68) In 2018,

Launched in 2015, the Supporting Entrepreneurship in Africa programme is a philanthropic initiative of the Tony Elumelu Foundation whose goal is to empower young African entrepreneurs.

The Tony Elumelu Foundation (TEF) was founded in 2010 in Nigeria by Tony Elumelu himself, an entrepreneur, philanthropist, and the chairman of United Bank for Africa. The foundation's objective is to participate in Africa's economic development by "empowering African entrepreneurs to create jobs on the continent".

The Tony Elumelu Foundation Entrepreneurship Programme (TEEP) is a ten year-long $100 million initiative committed by Tony Elumelu to provide training, mentorship, seed capital and networking to 10,000 African entrepreneurs at different stages of businesses they want to develop on the continent. Having participants from the 54 countries of Africa, the programme had 4,470 entrepreneurs by the beginning of 2019 and had given out $ 5,000 as non-refundable seed capital to each beneficiary. The goal of TEEP is to create at least 1 million jobs and contribute over $10 billion in revenue to African economies. Started in 2015, the TEEP launched its 5th edition in 2019.

The programme is led in lines with 'Africapitalism' the philosophy of its founder. The Afrocapitalist's vision emphasizes the role of the private sector in contributing to the prosperity of Africa. As such, beneficiaries of the programme are expected to create jobs and economic opportunities in their communities as well as to be agents of sustainable development. As the philosophical notion entails, the goal is to "create economic prosperity and social wealth. Africapitalism has the power to move all of Africa forward, and ultimately transform the lives of ordinary Africans in Africa". The programme is also executed with the support of the 7 pillars of the Tony Elumelu Entrepreneurship programme, which comprises the start-up enterprise toolkit, an online mentoring, online resource library, meet-ups, the TEF Entrepreneurship Forum, seed capital, and an alumni network.

Foreword

The Tony Elumelu Foundation Entrepreneurship Programme (TEEP)

Terhemen Agabo, a Master's student in Economics from the University of Ibadan (Nigeria)

Luc-Yaovi Kouassi, a Master's student in Political Sciences at the Panthéon Sorbonne Paris 1 University (France)

Two junior researchers were selected by IFRA-Nigeria to work on the project:

They were hosted within IFRA permises in Ibadan to prepare the fieldwork and analyse the data.

Their work was supervised and coordinated all along by IFRA's research associate, Clémentine Chazal and IFRA's Director, Dr. Elodie Apard.

The Tony Elumelu Foundation (TEF) provided a database of entrepreneurs as well as other logistical support for the work.

Luc-Yaovi Kouassi looked at the "Social and political impact of the Entrepreneurship Programme on the beneficiaries" from the perspective of political sciences. He explored the extent to which the Entrepreneurship Programme gives beneficiaries-entrepreneurs new resources they can convert into new social and political positions. His research is a comparative analysis between selected entrepreneurs in Nigeria, Cameroon and Côte d'Ivoire.

Terhemen Justine Agabo's research investigated the "Tony Elumelu Entrepreneurs and Living Standards in Rural Communities across Nigeria and Kenya" from an economic perspective. This study focused on the economic impact of the Tony Elumelu Foundation (TEF) Entrepreneurs operating in rural communities in Nigeria and Kenya looking at job creation, provision of good and services and the use of raw materials. It sought to uncover how they have improved the living standard of rural communities where they developed their businesses.

The two junior researchers conducted several months of fieldwork in the targeted countries and tackled the project with a different approach:

The two technical reports below highlight the findings of these studies. The content of these reports are the responsibility of their authors and that of IFRA-Nigeria; they do not reflect the opinion of TEF or AFD.

While the absolute number of people living in poverty, worldwide, has declined globally from 47% in 1990 to 22% in 2002, the number of poor people in sub-Saharan Africa has increased. [1] Furthermore, the bulk of the poor in sub-Saharan Africa live in rural communities where basic infrastructure is absent. [START_REF] Khan | Rural Poverty in Developing Countries: Issues and Policies, IMF working paper[END_REF] On the other hand, entrepreneurship can play an important role in mitigating the growth in poverty in the developing world as well as contributing to the achievement of the sustainable development goals. Micro Small and Medium Enterprises (MSMEs) have proved to be key drivers of employment, innovation and economic development in Africa, Asia, and Latin America. For instance, MSMEs account for 80 percent of jobs in developing countries.

Entrepreneurship is vital in the eradication of poverty especially in rural communities by providing decent jobs for individuals living in those communities, engaging the use of local raw materials owned by members of the local communities thereby creating markets for economic activities. [3] In other words, MSMEs generate social wealth and income through their ability to create new markets, new industries, new technologies, new jobs, and net increases in real productivity are thus able to reduce the poverty level. MSMEs have positive impacts in communities by mobilizing private investment into lagging areas and spreading the dividends of economic growth to individuals and places too often left behind. [START_REF]Promoting Entrepreneurship for Sustainable Development: A Selection Of Business Cases From the Empretec Network[END_REF] In Nigeria and Kenya, the promotion of MSMEs by the government has been going on since the 1960s when both countries gained independence. The governments of both countries recognize the importance of SMEs for their economy as they help to achieve the goal of eradicating unemployment and poverty. Both countries have adopted and reformed various SME support agencies and schemes such as the Small and Medium-scale Enterprises Development Agency of Going further, Entrepreneurship is essential in the achievement of certain of the development goals.

[7] For instance, the Sustainable Development Goal 1 is concerned with food security and MSMEs in sub-Saharan Africa play an important role in agriculture, which includes the production of food. Also, a good number of MSMEs have contributed to the production of clean energy, which is line with the seventh sustainable goal.

RESEARCH REPORT 2020 09 This study took place in Nigeria and Kenya. The chosen sample comprises selected entrepreneurs operating in rural communities, who were all beneficiaries of the Tony Elumelu Entrepreneurship Programme.

Interviews

Semi-structured interviews were used to obtain data from the respondents, who were a total of 40, including 20 entrepreneurs in rural communities (10 in each country) and 20 individuals belonging to these communities (10 in each country). The aim of the semi-directive interview method was to collect detailed information that would be useful in understanding the impact of the entrepreneurs in their communities.

The study having taken place in two countries permits a comparative analysis that will reveal varying and similar outcomes of entrepreneurs working in the two countries. Another interesting feature is that the two countries-Nigeria and Kenya-are economic leaders in the Western and Eastern Africa regions respectively, and that both countries have the highest number of entrepreneurs who have benefited from the Tony Elumelu Entrepreneurship Programme.

Quantitative and Qualitative data

Quantitative as well as qualitative data were sourced in the course of this study. The use of both kinds of data is warranted by the nature of the research, as it involves quantitative facts, some of which are best expressed quantitatively, while some detailed phenomena are best revealed using a qualitative medium.

Methodology

Study Location and Research Methodology

The fieldwork activities spanned over a period of six months (three months of intensive fieldwork and the remaining three for follow-up). The fieldwork in Kenya took place in June 2019, while that of Nigeria was spread between July and September.

A typical interview involved a visit to an entrepreneur's business site, a tour of the site, an interview lasting an average of 40 minutes with the entrepreneur, an employee and/or an individual living in the community. In some cases, an entrepreneur was not on site and had to be interviewed outside the community where his business was located (3 of such cases in both countries). Also, follow-up interviews were carried out in certain communities to verify or collect more information.

Fieldwork

Interview Locations Entrepreneur Among the many definitions of an entrepreneur, the definition by Joseph Schumpeter is adopted for this study. According to this renowned political economist, "an entrepreneur is one who takes up the task of reforming the pattern of production by exploiting an invention or, more generally, an untried technological possibility for producing a new commodity or producing an old one in a new way." Also, this study recognizes that very often Micro, Small, and Medium Scaled Enterprises (MSMEs) are used together with entrepreneurship. The two terms are not similar but related. This study defines MSMEs as employing a certain number of employees: Micro (1-9), Small (10-49), Medium (50 -99).

Rural Area

A rural area is defined in opposition to an urban area, as remote places, located within the countryside, composed of small villages and characterized by its low population.

Important definitions

20 Entrepreneurs = 345 jobs

Total employment created by selected Entrepreneurs

A total of 345 jobs [START_REF]African Union Commission and OECD[END_REF] have been created by the 20 entrepreneurs across the communities covered by this survey.

Impact of jobs on communities

The findings of the research show that entrepreneurs create jobs for members within host communities. The employment range is from 1 to 50 workers. This implies that some entrepreneurs create way less employment in their communities than others. However, discussions with community individuals show that the presence of jobs, even few, provide a source of livelihood that enables them to meet their needs and those of their families. Another notable feature is the fact that more than half of the jobs provided by entrepreneurs are temporary. Entrepreneurs explain this phenomenon as an outcome premised on cost avoidance. The temporary nature of the jobs raises the question of the livelihood sustenance as well as that of the quality of jobs, discussed in the next subsection.

Quality of jobs created

The quality of jobs also varies with company age, revenue, and location. Observations show that entrepreneurs whose companies are older afford to pay health insurance, pensions and provide loans for their employees. The same goes for companies with high earnings. Also, entrepreneurs in Kenya offer more quality jobs in terms of salaries and other benefits, such as health insurance and pensions. This is an outcome of the variation in the labor regulatory institutions in both countries. The quality of jobs goes a long way to determine the well-being of workers. Interactions with workers in enterprises with good working benefits show that such workers are able to afford further education while maintaining their jobs. This way, they are well-positioned for competitive salary compensations. As stated earlier, most of the jobs provided by entrepreneurs are temporary or seasonal. Given the seasonality of production in high employment generation sectors like agriculture, entrepreneurs avoid the high cost of

Findings

Impact of the TEEP on rural areas from selected businesses [START_REF]African Union Commission and OECD[END_REF] This figure represents the total job count in the enterprise branches visited in this survey. The complete figure of jobs created by the entrepreneurs across all branches of their business is reported in the section-Impact of TEEP on the ecosystem.

maintaining workers that will not have tasks to perform all year round. However, temporary jobs bring about job uncertainty on the part of the workers and, as observed during the field visits, several workers hang around business premises hoping to be called to work. Others choose to work for free rather than staying idle.

Firms in the agriculture and manufacturing sectors employ the highest number of people, as can be seen on the chart. These sectors involve labour-demanding processes, which is why they have been a good source of employment for rural dwellers.

However, businesses in a sector such as ICT, financial services, and construction have contributed the least to employment in the communities. ICT and financial services companies visited do not require a high amount of labour to carry out their businesses.

Construction capacities Liquid soap

Food; Potato chips, Banana, Palm oil, Food processing services: Rice processing Body cream ICT services

Impact of Goods and Services produced by Entrepreneurs in Rural Communities

Impact of goods and services

Very often, those who dwell in rural areas lack access to basic and non-basic goods and services that would make their life easier. Some of these goods and services include food, healthcare and quality education among others.

From the findings of the study, entrepreneurs have sometimes been able to meet the need for certain goods and services that were not available in these communities before they started their business. As a result, living conditions have improved for some individuals in some communities.

For instance, the production of energy-saving stoves by an entrepreneur in Gbazungu village in Kaduna state helps to save on the cost of preparing a meal, as the entrepreneur himself explains: "some people spend as much as 500 naira in cooking a meal, so if you have these stoves that reduce the cost of cooking this same meal by 80 percent so you are spending anything between 150 to 200 to cook the same meal so it saves money." Also, the availability of ICT training facilities in rural areas helps rural dwellers gain useful knowledge for employment and educational purposes.

The affordability issue

The availability of goods or services exists independently of its affordability. The findings of this study revealed that some goods produced in rural areas are not affordable to everyone; hence a good number of rural dwellers cannot afford them. One example is construction services, a highly expensive service that can only be accessed by the wealthy.

On the other hand, some goods are affordable by the rural dwellers. They include food products: palm oil, banana mixture, potato chips for example, and equipment such as cooking stoves. Among services, access to micro-credit appears particularly relevant for rural dwellers.

Goods and Services produced by the community

Another interesting finding is the observation that, in many cases, the entrepreneurs were involved in the production of goods and services that were already being produced by several households in the community. This occurrence was particularly popular for food products.

Rural areas in Africa often serve as food production centers. Therefore, the entrepreneurs who produce processed food have to sell the bulk of their products in markets outside the communities, especially in the nearest cities to the communities. As such, entrepreneurs who produce processed food do not significantly meet the food needs of their host rural communities, as almost every household is involved in producing one type of food or another.

Use of Local Raw materials by SMEs

Impact of the use of local raw materials by Entrepreneurs

From the findings of this research, the use of local raw materials by entrepreneurs provides an avenue for local people to earn income by selling to the entrepreneurs. It also leads to the establishment of businesses, mostly farms and mineral mining by the local people. These new businesses set up to meet the needs of entrepreneurs sometimes develop into a source of livelihood to the rural dwellers.

The use of local raw materials varies by sector. Businesses in construction, ICT, manufacturing and financial services do not depend significantly on local raw materials. The reason for the low use of local raw materials by companies in these sectors is due to the unavailability of relevant raw materials in rural communities. For instance, one of the SME owners who work in the textile industry says:

"We go to Nairobi to buy our materials. We do not have them in Karatina. So, we get them from Nairobi and make our products in Karatina and sell them there"

On the contrary, businesses in agriculture and manufacturing use local raw materials in a good quantity owing to the abundance of agricultural produce in rural communities. Some companies in the manufacturing sector also find several local raw materials useful in their production activities.

One entrepreneur who manufactures stoves explains the major constituent of his product: "We source clay form here, the sawdust there is a sawmill here, we use clay, we use kaolin, and kaolin is from here"

The use of local raw materials can keep rural dwellers engaged in economic activities in their areas.

Evolution and vision of selected businesses

The findings of the study reveal that some entrepreneurs are socially conscious and have shown a commitment to provide solutions to social, environmental, and economic challenges to their host communities. Some of the selected entrepreneurs founded businesses with the motivation to address social issues, while others who were driven into business by a clear profit motive. Several have, however, identified ways to contribute to the attainment of Sustainable Development Goals (SDGs) in their communities. Some selected cases that fit this description are narrated below:

Fighting the Felling of Shea trees in Kayama Community in Kwara State, Nigeria.

Adebola Ahijo founded the "Save the Shea initiative" to fight the felling of shea trees in Kayama community in North-Central Nigeria, felled trees being used for charcoal production there. Adebola, who is into the production of baby cream, uses Shea butter from the Shea trees as a major raw material. Also, the Shea trees have been a source of livelihood to the women in the community for more than five decades. Concerned that the sustainability of her business and the sustenance of the women's livelihood in the community were under threat, she launched the initiative. She mobilized local women, local leaders, and officials of national Agencies to act against the felling of Shea trees. As part of the campaign, Adebola is currently working on a Shea tree plantation to keep her business and ensure the sustenance of the local women's livelihood who work in the Shea plantations.

Ensuring Sustainable Food production in Idah Local government, Kogi State, Nigeria.

Eleojo Peters retired from a 15-year banking career for the sole purpose of establishing a grain-processing plant in her village in Idah local government.

Moved by the high rate of poverty in her community, Eleojo mobilized farmers to cultivate rice, sorghum, and other grains in high amounts. Also, she set up a riceprocessing factory at the center of the community. With the availability of modern machinery in the factory, the farmers can now produce more grain and have it properly processed, which makes it easy for storage, improves the quality of the grains and enhances its selling potential.

Healthy drinking water in Naivasha, Nakuru

County, Kenya. Brian Ayodi's cousin was diagnosed with Liver Fluorosis caused by high amounts of fluorine in his community's underground water. This ordeal drove Brian to think of a solution that would be accessible to the whole of Naivasha community. Brian came up with a water filter that reduces excessive amounts of fluoride in drinking water. He is currently working on improving the product for mass production.

Food security in Kenya.

Mwea County produces 80 percent of the rice in Kenya. Most of the rice farmers in the community use inorganic fertilizers which may improve crop fertility in the short term but damage the soil after some time. To mitigate this challenge, Samuel Riga, a Tony Elumelu entrepreneur, started a company that produces an organic fertilizer suitable for the soil of Kenya. The fertilizer from Safi Organic improves crop yield, preserves soil quality, and is more affordable. With the use of an organic fertilizer, sustainable food production is certain in Kenya.

A visit to Safi Organics, Mwea county, Kenya

Impact of the Tony Elumelu Foundation Entrepreneurship on the Business Ecosystem in Kenya and Nigeria

Further findings from the study show that the Tony Elumelu Foundation Entrepreneurship Programme has had a significant, albeit unequal, impact on entrepreneurs' businesses.

TEEP and Business Take Off

Out of the twenty businesses visited, five businesses, representing 25% of the sample, took off as a result of the training and especially the funding provided by the Tony Elumelu Foundation Entrepreneurship Programme.

Business Growth/Value Addition

Out of twenty businesses, two belonging to the agricultural sector have moved from crop production to production of food products. This way, more revenue and employment were recorded.

Employment

After benefiting from the Tony Elumelu Foundation Entrepreneurship Programme, the total number of jobs provided by entrepreneurs increased from 132 (before TEEP) to 770 jobs (after TEEP). [START_REF] Elumelu | [END_REF] Revenue Also, the total revenue generated by the entrepreneurs before TEEP stood at $690,478. However, this figure increased to $3,785,878 after TEEP.

[9] Total number of jobs include both temporary and permanent jobs created by the entrepreneurs. In some cases, jobs of a particular firm included those in rural and urban communities.

Recommendations

The majority of the jobs provided by entrepreneurs in this study are temporary in nature, due to the operational cost and finance-related reasons. Entrepreneurs should be supported with finance and infrastructure to grow their businesses so they can offer quality jobs for individuals in their communities.

The manufacturing and agricultural sectors provided the highest number of jobs for communities. Entrepreneurs in these sectors should be given greater support so as to curtail unemployment in communities.

Meeting Regulatory standards remains a cost-intensive project for entrepreneurs. There is a need to subsidize the regularization process for entrepreneurs, to enable them to sell their products in various markets across their countries and in foreign countries.

The level of technological innovation in Africa remains at an infant stage. Entrepreneurs working on indigenous technology should be linked with experts from technologically advanced countries to enable them to develop fully usable products.

Businesses contribute to the attainment of Sustainable Development Goals (SDGs) in rural communities; contributing to the enhancement of local people's livelihood should be given greater support and be prioritized.

Introduction

Entrepreneurship and entrepreneurs have been increasingly perceived by regional and international institutions as new means to reduce poverty and create wealth in Africa. Published by the African Development Bank (AfDB), the United Nations Development Program (UNDP) and the Organization of Economic Cooperation and Development (OECD), the 2017 Africa Economic Outlook, whose special yearly focus was on entrepreneurship, shared this point of view: thanks to an "entrepreneurial culture" "spirit" and "dynamic private sectors", Africa has great potential to thrive, but only if, amongst other preconditions, "investment in human capital", i.e. providing entrepreneurs with enough funding, skills and education, is seriously undertaken [1]. That is why several programmes aimed at supporting successful-entrepreneurs-to-be have been started all over the continent. This idea is not new, however. After the disappointing results of the structural adjustment policies leading to the dramatic collapse of main economic, political, and social indicators of African countries [START_REF] Coussy | Etats africains, programmes d'ajustement et consensus de Washington[END_REF][START_REF] Herbst | The Structural Adjusment of Politics in Africa[END_REF]), the lens of the international developers changed at the end of the 1980s. While their efforts had so far been aimed towards state "reform," the private sector-and more specifically entrepreneurship-began to rise as a new means to tackle socioeconomic challenges in Africa. The Marsden report, published on behalf of the World Bank, demonstrates the viability of African entrepreneurship, the latter being a source of GDP growth, able to reduce unemployment and constituting a catalysing tool of "social progress" thanks to the innovative activities supposedly put in place [START_REF] Marsden | African Entrepreneurs: Pioneers of Development[END_REF]). The novelty, the trend that one can observe today, lies probably more in the fact that pro entrepreneurship programmes are now put in place by the African private sector, such as the Confédération Générale des Entreprises de Côte d'Ivoire (CEGCI) or the Tony Elumelu Entrepreneurship Programme (TEEP). The latter best illustrates the social capacities of entrepreneurship; as the philosophical notion of 'Africapitalism' entails, the goal is to "create economic prosperity and social wealth. Africapitalism has the power to move all of Africa forward, and ultimately transform the lives of ordinary Africans in Africa" (TEF 2018). Little is known about the political and social consequences on individuals who passed the extremely competitive selection. As a matter of fact, in 2018, only 1,250 individuals were selected out of 151,692 applicants. But who are those persons who benefited the programme? What have they become after the annual programme cycle? Those questions are important with regard to the historicity of capitalism and entrepreneurs in Africa. Historians have shown that, after the decolonisation of Africa that mainly took place in the 1960s, states have been perceived by the population as the main driver of development and enhancement of living standards [START_REF] Cooper | Africa since 1940: The Past of the Present[END_REF]. Moreover, dominant social groups have been able to thrive thanks to straddling political positions-or closely related to the state-and economic positions [START_REF] Bayart | The State in Africa. The politics of the belly[END_REF]) because of the influence of developmental colonialism on the state formation [START_REF] Cooper | Africa since 1940: The Past of the Present[END_REF]. That is why some scholars theorised the notion of "political-entrepreneur", describing a civil servant, a politician and/or a businessperson (ideally the three in one) straddling economic and political fields, making use of resources acquired in one field to reinforce his/her position in the other field [START_REF] Médard | Le "Big Man" en Afrique : Esquisse d'analyse du politicienentrepreneur[END_REF]). The "normal cursus", or the regular trajectory, which had been described is the shift from politics to economics. A reverse trajectory of "entrepreneur-politician" also exists, albeit statistically rare. In other words, states-or politics-in Africa constituted the matrix of capitalistic accumulation because of the weak autonomy of the economic field. At the same time, the TEF aims at fostering the "next generation of African entrepreneurs" who, one can assume, should contrast with the previous generation of entrepreneurs. Therefore, this report proposes to analyse the relationship between members of this so-called next generation of African entrepreneurs and the state, working on the assumption that this particular aspect would vary between those two generations. This assumption seems credible, especially since African states (at least some of them) are only one partner of the programme amongst others (world states, developmental organisations, ventures, etc.).

Theoretical framework

Methodology

To analyse the relationship between TEF entrepreneurs and their state, a qualitative approach was adopted: semi-structured interviews were conducted with a relatively small number of individuals (n = 39) from three countries (Côte d'Ivoire, Cameroon, Nigeria) in order to take into account an extensive set of variables. Considering how sociologists and political scientists addressed the topic previously, meetings with respondents were based on one main biographical, semi-structured interview, of a length comprised between 90 minutes and four hours (depending on the respondent's schedule) at a location chosen by each respondent. This main biographical interview, followed when possible by observations at the respondent's place of business, is the only interview displayed by the following charts, but other unstructured interviews, always written down in my field notebook, did occur. One third of the meetings were reiterated one or two times.

Interestingly, a distinction can be made between two types of respondents, leading to different relationship to the inquiry. Indeed, the common professional identity that the respondents share cannot blur for a long time the heterogeneity of their social situation. A minority of entrepreneurs enjoys and emphasizes some attributes of success, such as suit-and-tie, assiduous and working employees, spacious office, etc. In other contexts, such individuals were called "imposing figures" [START_REF] Laurens | Why' and 'How' to Ask Irritating Questions? Reflections on the Recurring Dilemmas Posed by Interviews with 'Imposing Figures[END_REF]. In terms of personal income and business revenue, they are the most successful individuals. Such "imposing figures" tend to have the upper hand during interviews, bending questions to their advantage and presenting themselves from the most conventional, emphatic way possible. This can be explained by the fact that most of them are used to media coverage: Ange-Pierre A. was described as a the "African golden nugget" and interviewed by a French national TV. They also tend to be more open at first-"imposing figures" are usually the first to respond to contact requests by email-but interviews appear to be shorter. It is also more difficult to meet them again after the first interview, as they argue that they are overscheduled. On the contrary, there are respondents who are entangled in less stable situation, or even stopped running their business. Sometimes, ashamed by their situation they do not want to show what could appear to be failures -especially to a researcher sent by the TEF and the AFD. Respondents falling in this category made initial contacts tougher (not responding, postponing rendezvous…), but turned out to be more open to additional interviews.

Entrepreneurs were reached by email. Their addresses were obtained via the TEF database or through previously interviewed beneficiaries. Relying on the personal knowledge of one another helped circumventing the trap of the methodological bias of reaching out only the most successful beneficiaries. Despite this solution, only a fraction of entrepreneurs contacted actually replied. In Côte d'Ivoire, of the 68 entrepreneurs contacted by one mean or one other, only 24 replied -whether favourably or not. 13 entrepreneurs were met. In Cameroon, of the 47 entrepreneurs contacted, only 22 replied. 12 of them were met. In Nigeria, 78 were contacted and 31 replied [START_REF] Khan | Rural Poverty in Developing Countries: Issues and Policies, IMF working paper[END_REF]. 14 interviews were conducted. The discrepancy between those who replied and those who were interviewed can be explained by several reasons: some entrepreneurs moved to other parts of the country, were not available at the moment, kept postponing the appointment, stopped answering, etc.

The fieldwork took place in Abidjan (Côte d'Ivoire), Yaoundé and Douala (Cameroon), and Lagos and Ibadan (Nigeria). These choices are the result of arbitration between feasibility and representativity. Indeed, most of TEF beneficiaries develop their activities in urban areas. As a matter of fact, in 2018, 71,8% of them established their ventures in cities. Moreover, as the following maps show, a significant amount of TEF are located in the South West of Nigeria, in Abidjan, in Douala and Yaoundé. This can be explained by the economic dynamism of these areas: Abidjan represents almost 19% of the total population of the country, while accounting for 28.7% of the national revenue generated by households [START_REF] Nallet | De nouveaux lieux de consommation pour de nouvelles classes moyennes ?[END_REF]). Douala, the economic centre of Cameroon, produces most of the national wealth and makes up almost half of private employments, while Yaoundé is the capital city where the tertiary sector prevails [START_REF] Mintoogue | L'épargne des classes moyennes au Cameroun. Une économie en marge du système bancaire[END_REF]. In Nigeria, South West, especially the Lagos-Ibadan corridor, concentrates important economic activities, home of many ICT, financial services and manufacturing firms which makes it the area with the highest GDP per capita (Bloch et al. 2015). The geographies of TEF beneficiaries vary a lot between those three countries. In Nigeria, although clearly uneven, the distribution of entrepreneurs is nation-wide. In Côte d'Ivoire, practically all entrepreneurs are located in Abidjan. Two entrepreneurs met, who registered themselves in other parts of the country to the TEF, have moved to the city, suggesting a powerful attractiveness of Abidjan for national entrepreneurs. In Cameroon, four regions set themselves apart in terms of TEF beneficiaries: the Littoral region (Douala), the Centre region (Yaoundé), the South-West (Buea) and the North-West (Bamenda). Despite the importance of the last two regions in terms of beneficiaries and entrepreneurial ecosystem (the famous Silicon Mountain, a ITC hub, is located in Buea), it was impossible to visit them because of the current Anglophone crisis (International Crisis Group 2019). Nonetheless, two TEF beneficiaries from the South-West and the North-West were interviewed after they moved out of these regions to Yaoundé and Douala. 2). Respondent's age varies between 23 and 60 years old (average: 32). The level of education is high, as everyone (except two respondents) obtained at least a college degree.

One could qualify the respondents as "small entrepreneurs" [START_REF] Fauré | Petits entrepreneurs de Côte-d'Ivoire: des professionnels en mal de développement[END_REF]) because of their company annual revenue and the number of their employees. Taking into account only the revenue of the company they applied to the TEEP for (excluding therefore three cases of business bankruptcy and other that respondents may have), it goes from 1 to 350 million francs a year (N216,618,500). Four companies (one in Cameroon, three in Côte d'Ivoire) exceed 100 million francs a year (N61 891 000). The majority makes a revenue between 10 (N6,189,100) and 40 million francs (N24,756,400), depending on the years. Nigerian business' turnovers inquired are also within this range. In terms of manpower, only three companies in the industrial sector, hire more than 12 employees, while half of the compagnies do not have employees in Côte d'Ivoire and in Cameroon. The rest hire between two and eleven employees. In Nigeria, no business hires more than seven workers.

Structure

First, the report analyses how respondents started their businesses, how they began to accumulate capitals in order to assess whether or not their trajectories correspond to those of politician-entrepreneurs. Indeed, some previous state-run entrepreneurship programmes mainly benefited politician-entrepreneurs. For example, in Côte d'Ivoire, in the 1980s, the government tried to implement such mechanisms to support local entrepreneurs but failed, as most beneficiaries were actually chosen on the basis of their proximity to political big men rather than their ability to start and lead a prosperous business [START_REF] Fauré | Petits entrepreneurs de Côte-d'Ivoire: des professionnels en mal de développement[END_REF]. Then, the report proposes to describe the potential interventions of TEF beneficiaries in the political field.

[3] On the first four editions, 32% of beneficiaries were women.

Findings

To compare TEF beneficiaries to the category of "politician-entrepreneurs," it is crucial to concretely examine how respondents launched into entrepreneurship and particularly how they accumulated sufficient funding, capital, in order to sustain their project Fauré distinguished different sources of funding: self-funding, loans (commercial banks, microcredit institutions, tontines) and gifts, especially coming from the familial sphere (Fauré 1994, 186-95).

Every respondent insisted on the necessity of injecting their own funds, usually accumulated through their own employment, to launch their business. For almost everyone, this is the main source of capital they have. This self-funding process is accompanied by a thorough saving behaviour.

Thierry H., in Cameroon, is employed by a furniture craftsman in a workshop but tensions arise about management after he received a training on SMEs funded by the Institut Européen de Coopération et de Développement. As he felt professionally restricted because of the business executive's poor managerial skills, he planned to establish his own workshop. By 2013, without any income except his job, he sent his family back to his village, in the West region where his son and his wife were supported by the community while he kept working for his employer. This strategy enables him to double his annual saving capacity, growing from 100,000 (N61 891) to 200,000 francs (N123 782).

Respondents also explain how they "make sacrifices", diverting incomes from consumption to their entrepreneurial project. However, one should stress a strong inequality in what appears to be a common process. On the one hand, a clear minority of respondents "come into the business through the front door" (Miaffo and Warnier 1993), i.e. thanks to formal, stable and well-paid employment.

Ange-Pierre A., prior to his successful entrepreneurial journey, worked as an IT specialist for a transnational firm in Côte d'Ivoire at the end of the 1990's. He was rapidly promoted and sent to work to France for two years. Back to Côte d'Ivoire, he was hired as head of the IT department of Orange, a French multinational telecommunications corporation, earning around 7,000 euros a month (N2,841,860). Thanks to his connection to French telecommunications staff and his

Processes of accumulation disconnected from politics

connection to French telecommunications staff and his savings, he was able to start a viable tourist business in Côte d'Ivoire. This flourishing business would then, in turn, support his next business idea which was submitted to the TEF seven years later. This category of entrepreneurs -they generally correspond to the "imposing figures" aforementioned mentioned-are older than their counterparts and successfully obtained formal jobs -always in the private sectorafter they graduated.

On the contrary, the vast majority of the respondents "come into business through the back door". They also fund their "TEF business" from their own employment. However, the types of jobs they have differ greatly from what those who came through the front door enjoy. Local expressions in the three countries describe differently the same reality: "gombos" in Côte d'Ivoire, "gombos" [START_REF]Promoting Entrepreneurship for Sustainable Development: A Selection Of Business Cases From the Empretec Network[END_REF] or "jonglage"[5] in Cameroon and "hustle" in Nigeria. They constantly refer to unstable, informal, precarious jobs. In Cameroon, Christelle E., after winning three entrepreneurship competitions including the TEEP, funded herself mainly from her "gombo", buying beauty products abroad and reselling them locally through Facebook until she felt "solid" enough to remunerate herself from her waste management company that she just founded.

It appears that "hustles" are sometimes linked to the business idea submitted to the TEF. In Côte d'Ivoire, Providence K. was selling yam and plantain bought from farmers in the North-West until he had the idea of transforming them into crisps. Entrepreneurship programmes, including the TEEP, therefore helped him formalise his "gombo". It is worth paying attention to what become "hustles" after the support of the TEEP. Hustles may be abandoned as soon as one get awarded. After graduation in ITC, several poorly-paid internships and "way too much" ("beaucoup beaucoup trop") job refusals, André H. started repairing computers informally aiming at capitalising enough to start a security company. Despite his efforts, he did not achieve to elaborate a practical project nor to save enough money. He then focused on another project which he applied in 2017 to the TEF for. Even though he had no other incomes at that time, he abandoned his "gombo" because of how time-consuming this activity was and of the instability of incomes. On the contrary, those activities may remain on the side of the business they applied to the TEEP for, but can also be assimilated into it. Jérémie S., Ivoirian, participate [START_REF]Promoting Entrepreneurship for Sustainable Development: A Selection Of Business Cases From the Empretec Network[END_REF] Literally "okra" -staple food in the three countries and beyond-meaning an informal activity sustaining someone on a daily basis.

[5] Literally "juggling", indicating the adverse and precarious situation of this kind of jobs, but also the agility and astuteness required to do them.

d in several several entrepreneurship programmes and his project has been rewarded multiple times.Yet, his project, developed since 2011, has not been implemented. The company he administratively created to receive the TEEP funding is mainly used for other activities, such as importing electronic devices from China to Côte d'Ivoire. More generally, in every category, there is a clear tendency of pluri-activity, or multiactivity. Whether trough hustles or multiple registered ventures, respondents seem to maintain co-dependent activities, even though one of them, usually the one they submitted to the TEF, is seen as a promise of success. These activities contribute to generate a relative flow of income, despite the inherent precariousness of informal works. This underlines an interesting trend from a policy-making point of view: if informalization of the African economies must be tackled, notably because it hampers poverty reduction [START_REF] Bhorat | The Pursuit of Lang-Run Economic Growth in Africa. An Overview of Key Challenges[END_REF], one must realize that formalisation is a long process. The TEEP certainly pushed towards the formalisation of businesses, but respondents may choose to keep an extensive amount of their activities informal because it provides flexibility and a safety net.

Consequently, entrepreneurs who "came in through the front door" are more likely to contract loans. More precisely, not all entrepreneurs who worked and saved money from formal and well-paid employments use loans; but almost all loan debtors (except two in Côte d'Ivoire and two in Nigeria) are entrepreneurs who "came in through the front door". Six entrepreneurs contracted a loan from a public or a private bank, going from 3 million (N1,856,730) to 52 million francs (N32,183,320). In Nigeria, seven respondents contracted loans (up to 6 million nairas), including five who "came in through the front door". As mentioned elsewhere [START_REF] Ha | Sutien Des Écosystèmes d'innovation Numérique : Quel Rôle Pour Les Institutions de Développement ?[END_REF]), high interest rates and required collateral play as deterrent. This corroborates an observation made two decades ago about Cameroonian entrepreneurs (Miaffo and Warnier 1993). At that time, their alternative consisted in participating in tontines (equivalent to Esusu or Osusu in Nigeria).

[6] Among respondents, this mechanism is very rare. Only two entrepreneurs in Cameroon used it with more or less success (Léonard N. successful raised 20 million francs (N12,378,200) through the elders' tontine of his village association, while Thierry H. only managed to get 200,000 francs -N123,782). In Nigeria, only one entrepreneur sees tontine, Esusu, as a mean to access additional funding through farmers' associations. The small roles played by tontines in respondent's pathways of accumulation is surprising, considering

[6] Associations or clubs, mainly informal, organised on the basis of social links (friendship, profession, gender, age…) where every member pays a contribution regularly, to raise funds. Working on a rotating manner, every member is debtor then creditor how they had been praised in the scientific literature as well-suited indigenous tools capable of fostering African entrepreneurship [START_REF] Rozas | Les tontines favorisent-elles la performance des entreprises au Cameroun ?[END_REF]. Is the reason to be found in the generation to which the respondents belong? This is one factor, especially in Nigeria where tontines are seen as "something from the past". Another factor also lies in the amount of money that one can expect to raise through tontines, seen as not high enough.

In this context of scarce sources of funding, which are likely to help consolidate a viable business, entrepreneurship programmes are seen by most respondents as a way to get more capital.

See appendix: Summary of entrepreneurship programmes Cameroonian et Ivoirian respondents went through

Workers sorting out recyclable cardboards in Abidjan

In Cameroon and Nigeria, only three respondents limited their programme participation to the TEF alone. However, this strategy is not flawless: some TEF entrepreneurs fiercely criticise these "socalled" entrepreneurs who follow it, labelling them "award collectors". They fear that those "collectors" will capture available funding opportunities for their owngreed instead of creating jobs and put in place innovation. Thierry H., TEF representative for Alumnus in Cameroon, would like to see a stricter control over time on beneficiaries, as many of them ignore his phone calls. However, one should not take for granted this point of view, which comes from a socially situated group. Respondents who are the most critical of the "award collectors" appear to be those who built up a functional, operational business, often after saving from formal jobs. Entrepreneurial awards and certificates do more than simply grand access to additional funding: their accumulation seems to be a tool used by the less endowed to progress as active members of their society. Indeed, their incapacity to reach positions that lead to what is considered to be success -positions usually related to jobs in the public sector in Cameroon and in Côte d'Ivoire [START_REF] Banégas | Figures de la réussite et imaginaires politiques[END_REF] -fundamentally explains most of their itineraries. Interestingly, this phenomena of accumulating entrepreneurial awards and certificates is not that noticeable among Nigerian respondents. Even though they are looking for new opportunities, less entrepreneurs seem to be engaged on the successful path of entrepreneurial awards accumulation. One hypothesis could be the existence of rigged programmes, as the controversy around YouWin Program shows us [7]. This situation would be comparable to what existed in Côte d'Ivoire during the 1990s, when entrepreneurship programmes put in place by the government were engulfed by clients of political big men [START_REF] Fauré | Petits entrepreneurs de Côte-d'Ivoire: des professionnels en mal de développement[END_REF].

Morty R, an example of a TEF Alumni who "has fallen" ("qui est tombé")

Morty R. is an Ivoirian entrepreneur who dedicated his professional life to developing ideas into business. After failing to pursue higher education (he only has an undergraduate degree, a Brevet de Technicien supérieur) and transforming an internship into a regular job, he increased in 2006 the size of his former student informal occupation, a computer maintenance company. In its heyday, this activity generated a turnover of 2,000,000 CFA francs (N1,236,000) a month and hired a staff of three permanent employees. He also diversified his activities by launching a communication and consulting firm and an ITC school. But in 2010-2011, everything he has built collapsed. First, because his employees "run off with the takings" ("sont partis avec la caisse") to start a competing business in the same sector. Second, his offices were damaged and plundered during the electoral crisis. As he did not formalised his business, he was not insured and lost "all his savings". In order to bounce back, Morty R. worked with his cousins on a green coal project, while offering entrepreneurship trainings and repairing computers to survive on a daily basis. He applied for the TEEP in 2015 with this project and also received support from a national scheme in the form of a loan. He was actually never able to start the production of the green coal. According to him, the capital needed to start his business is near 100 000 euros (N40,422,574), a sum a lot higher than what African entrepreneurship programmes offer in terms of funding, which are more suited for service-oriented companies.

In the case of TEEP beneficiaries, the process of accumulation which led to the business creation is clearly different from what can be expected from "politician-entrepreneurs". Their introduction into formal business is realised from the informal sector or the formal private sector. However, does this mean that they remain separate from politics? Are there trajectories of "entrepreneurs-politicians"?

Respondents draw a clear distinction between business and politics, as if those terms were magnets repelling each other. At first glance, TEF beneficiaries interviewed seem apolitical: only one respondent stated that he was a member of a party (MRC in Cameroon, opposition). A couple of entrepreneurs used to be activists (André H. was a Young Patriot during the Ivoirian crisis in 2010-2011) but they now perceive this period as definitely over. As the Ivoirian Alexandre N. puts it, "in our tropics, business and politics don't mix well". Ivoirians and Cameroonian political configurations have different historicities [START_REF] Akindès | On ne mange pas les ponts et le goudron » : les sentiers sinueux d'une sortie de crise en Côte d'Ivoire[END_REF][START_REF] Amougou | Ambition développementaliste, État stationnaire et extraversion au Cameroun de Paul Biya. Le projet de construction du port autonome de Kribi[END_REF] but one common point lies in the proximity between the regime and wealthy businessmen. For example, the latter, through employers' unions -for example CGECI in Côte d'Ivoire -organise competitions for young entrepreneurs which are criticised for being rigged (such criticisms are more recurrent in Cameroon, which makes sense regarding the construction of a hegemonic bloc through the CPDM). Compared to other fieldwork, this situation may appear singular because Ngouyamsa underlined that, in Cameroon, the cost of remaining neutral or worse-getting involved with the political opposition-represents a powerful incentive for allegiance towards the regime, such as becoming a member of the ruling party or displaying conspicuously signs of support towards the ruling party. [START_REF] Ngouyamsa | Militantisme Politique et Entrepreneuriat :: relations de dépendance et enjeux dans le contexte Camerounais[END_REF].

How can one explain this discrepancy? One reason is the digitalisation of the respondent's activities, which makes harder the capacity of local barons or greater authorities to exert pressure on them, unlike Ngouyamsa's respondents who are petty shopkeepers. This does not mean that they avoid all political disturbance: in 2017, the Internet was shut down in the South-West and North-West region, as part of a governmental countermeasure in a context of a growing "Anglophone crisis." From a macroeconomic perspective, this was a massive blunder : the net loss due to this crisis is evaluated at 270 billion francs -167 billion naira (GICAM 2018). From a microsocial perspective, this led Desmond T., a TEF beneficiary established in Bamenda, North-West region, to stop most of his activities for 8 months and lay off his six employees. He then moved to Yaoundé where he focused on a "gombo"-importation of juices 2. Towards "entrepreneur-politicians"?

from Egypt-that he started after he completed his studies in China in late 2013.

Soy cooking

Owner's daughter saoking soy before transforming it, Abidjan However, respondents do not ignore political matters, as shown by the attempts to create organisations gathering TEF entrepreneurs on a national basis. Although few are individually members of trade unions (Thierry H. has been appointed secretary of an association of rattan producers, thanks to the new notoriety owed to his selection by the TEF), some respondents, belonging to the two first cohorts, tried to formalise formal groups that could represent to the authorities of their country. In Cameroon, a group of five beneficiaries of the first cohort registered an association which would regroup every TEF Alumnus. The purpose was to seek support from the authorities -in terms of subsidies and publicity -after realising that their own country, contrary to the Nigerian embassy to Cameroon, which invited them after their return from the Bootcamp, had no interest in them. Yet, three entrepreneurs from this group, who were interviewed, regret that the TEF did not back their initiative. With no legal existence as a group, they argue, TEF entrepreneurs would be invisible in Cameroon. In Côte d'Ivoire, a broader group of beneficiaries also tried to bring together their counterparts in 2017 but a controversy soon arose: one of them attempted to grant access to an accelerator in Abidjan in exchange for money. This led to the scattering of the group. In both cases, TEF beneficiaries do not form a unite group and consequently regular meetups that gather a fair amount of them do not take place. This situation is even more notable in Cameroon where the sociolinguistic division between Anglophones and Francophones, besides the fragmentation of beneficiaries all over 4 areas, makes interactions more difficult.

The last two trends that may push TEF beneficiaries to politics are the co-optation by big men through other entrepreneurship programmes and indirect intervention in the political competition. Co-optation takes place in entrepreneurship programmes organised at the national level. Frequently underthe patronage of a big man (an "entrepreneur-politician" or a "politicianentrepreneur"), some entrepreneurs explain that they had been under pressure to support them. Several respondents state that politicians asked them to publicly promoting them. Jérémie S., for instance, is regularly to be seen ("il m'affiche") with an influential counsellor to the Prime Minister after they met during an entrepreneurship event. Although Jérémie S. acknowledge that fact that the minister "is using [him]", he cannot break this relationship. On the one hand, it plays as a passport, enabling him to participate to more events, which makes sense in regard to strategy of certificates, awards and funding. On the other hand, he feels compelled to maintain relationship because of the fear that "they can end everything [he] has done" ("ces gens-là peuvent terminer tout ce que j'ai construis jusqu'à là"). This is another aspect of the domination of the political field over the economic field. Support from the political field is indeed ambivalent to businesspersons. It is not necessarily possible, nor a priority, to objectively establish whether there has been a "project theft" from a sociological perspective. The fact that accusations and rumours of "project theft" circulate shows, on the one hand, that the cycle of accumulation of entrepreneurship programmes can go hand in hand with negative effects for applicants and beneficiaries. On the other hand, this phenomenon illustrates well the power relations that are at work in the entrepreneurial worlds. Behind the opportunities that this type of programmes can effectively offer, there are series of big men, and more generally transnational firms having a social, financial and technical base capable of exerting on small entrepreneurs strong pressures in a more or less legal and a more or less acceptable manner according to the respondents. Last but not least, if one takes gender into account, co-optation may take different forms. In the case of Christelle E., a young Cameroonian entrepreneur, a "senior governmental official" conditioned his "protection" to "sexual favours".

Indirect intervention in the political competition is even more rare. Even though respondents emphasize the fact that business and politics are two different fields that should not collide for the greater good of both, one can notice collisions. While he claimed to preserve a political neutrality, Parfait E. "helped" two candidates by funding them and put at their service his database -for which he had been funded. Thanks to his help, candidates had the capacity to "remind" voters for whom to vote during the election day. This kind of interferences between politics and business are tricky to investigate, as social norms in which respondents are entangled lead to euphemise them. For instance, Parfait E. did not help politicians as such, but "relatives". Therefore, the objective reasons that pushed him to behave this way, are blurred by justifications of selflessness and family solidarity. Is this only a coincidence that he financially supported a politician when he was thinking of diversify his investments by investing in real estate? Beyond the topic of (un)legitimate means of enrichment for entrepreneurs, this also raises the ethical question of the use of innovations supported by the TEF.

In conclusion, the relationship between selected beneficiaries of the TEEP and politics is far from corresponding to the model of the "politician-entrepreneur" theorised in the 1980-90s. As the TEF did not select this profile of entrepreneurs, respondents tried to develop their business without straddling different fields. This should lead to rethink the relationship between political and economic fields in Africa. The fact remains that respondents are "small" or "young" entrepreneurs, who have just taken steps in their entrepreneurial journey. To what extent will they be able to remain independent from politics? Is success in business, i.e. on the long run, compatible with neutrality? As the case of Parfait E. shows us, it is maybe not particular ethics or a well-understood cost-benefit analyses that explain this situation of non-straddling, but maybe rather their current size and resources.

Recommendations

No respondents were able to take the optional loan of 5,000 dollars. Even though a minority would have done so, a handful of entrepreneurs considered this interesting.

Considering the close relationship between the advent of entrepreneurs and politics, to provide a code of good conduct from Africapitalists. This could give a content to this concept [START_REF] Amaeshi | Africapitalism: A Management Idea for Business in Africa?[END_REF], as well as a safeguard for potential scandals.

To take into account the cycles of participation of awards/programmes. This could be done through developing synergies with other incubators, such as Jokkolabs, Akedewa, W Hub, etc. [START_REF]African Union Commission and OECD[END_REF] in order to allow African entrepreneurs to blossom. The TEF could try to seek consistency between those different schemes in terms of knowledge and assets. Another option could be to increase the amount of funding given, as one main factor influencing trajectories is the limited amount of financial resources.

To enhance collaboration between entrepreneurs and national authorities. This could embody Tony Elumelu's philosophy: "Africa's renaissance lies in the confluence of the right business and political action". [START_REF] Elumelu | [END_REF] This could be done by giving more autonomy to TEF groups in their local/national ecosystem. In Cameroon, as puts the "president" of the local TEF group, "without any legal existence, you are invisible to the authorities". So far, such groups were organised in a top-down approach, such as the Foundation overseeing elections. Autonomy might help entrepreneurs, as a trained collective, to positively influence their ecosystem, introducing needed changes to make the private sector thrive in accordance with their local constraints. There is however a risk of abuse, which could be avoided by providing strict guidelines and instituting an accountable and elected board at the top of the group.

2015: PASRES

[ 5 ]

 5 Small and Medium Enterprises Development Agency of Nigeria (SMEDAN). (2013). Smedan and National Bureau of Statistics collaborative survey: Selected findings [6] Kenya National Bureau of Statistics. (2016). Micro, Small and Medium Establishments Basic Report 2016.Nairobi: Kenya National Bureau of Statistics [7] Promoting Entrepreneurship for Sustainable Development: A Selection Of Business Cases From the Empretec Network, UNTCAD 2017 A visit to Nenu stoves, Gbazumu village, Kaduna state

  Goods and services produced by entrepreneurs who participated in this study:

[ 1 ]

 1 African Development Bank, Organisation for Economic Cooperation and Development, and United Nations Development Programme 2017.

Figure 1 :

 1 Figure 1: Chart of interviews conducted in Côte d'Ivoire

Figure 2 :Figure 3 :Figure 4 :Figure 2 :Figure 4 :

 23424 Figure 2: Number of TEF beneficiaries by district in Côte d'Ivoire

Introduction Introduction: the role of entrepreneurship in boosting livelihoods in rural communities

  Nigeria (SMEDAN) for Nigeria.[5] In Kenya, the government rolled out the Assistance to Micro and Small Enterprises Programme (ASMEP) in 2007 to create a favorable business environment for the growth of SMEs in the country. SMEs have been strategically featured in the development plans of the two countries over the years.[6]

  Others applied or successfully accumulated entrepreneurship certifications. In Côte d'Ivoire for example, Mohammed B. went through TechnoServe (2012), the Prix d'excellence du president de la République de Côte d'Ivoire (2014), SEED Stanford Transformation Programme Accra (2014), Entrepreneurship Award Francophonie 35.35 (2016), Award of the Best Young Entrepreneur CGECI, United Nations Programme for industries in Côte d'Ivoire (2018). Using the same strategy, Jérémie S. was able to raise 60 million francs (N37,134,600) over four years.

  The fair density of local entrepreneurship programmes that one can observe especially in Côte d'Ivoire comes with what it is viewed as menace of "projects thief" ("vol de projets"). In a weak position vis-à-vis promotors (multinational companies, national administration…), participants are unarmed in case of malignant use of their efforts. Parfait E. is often mentioned by other respondents as the best example of this phenomenon. After winning competitions abroad for an innovative smartphone application in 2012, he is contacted by Orange Côte d'Ivoire, which offered him a job. He declined it, but kept applyingand winning -entrepreneurship programmes, in France and in Côte d'Ivoire. By late 2013, he noticed that Orange Senegal was developing a similar project. While Parfait E. was trying to clear up the situation, Orange -in partnership with the state -had been putting in place the same project in Côte d'Ivoire. Therefore, Parfait E. found himself in the paradoxical situation of an award-winning entrepreneur for a project that is simultaneously implemented without him.

  Award Carrefour des Possibles Afrique Fondation Internet Nouvelle Génération 2012 2014: 100 innovation pour un développement durable laureate, national d'excellence du Ministère de l'Economie Numérique et de la Poste 2014 Award, Orange Fab start up Award, Fire Africa Award 2014, Concours Start-up Investir en Côte d'Ivoire CEPICI Award 2014 2015: TEEP, Premier prix de l'innovation de l'IT Forum Awards, Jury Special Prize Abidjan Start-up weekend ; Nest'Up laureate of Forum mondial de la langue française en Belgique 2016: West Africa HIIL Innovation Award 2017: Prix d'Excellence PRCI, Third Pize meilleure contribution à la Vulgarisation de usages du numérique PRCI, Special Prize Agrihack

	Entrepreneurship Programmes of respondents (per country)
	Côte d'Ivoire
	Jérémie S.	Award Swiss Centre for	2012: Parfait E.
	Scientific Research, Best Young inventors
	laureate African Innoviation Foundation
	2016: Smart Capital Award Business Plan
	Competition de la CGECI
	2017: Best Ivoirian Start up African
	Rethink Awards, Falling Walls Lab Berlin
	laureate, First Prize concours de
	l'innovation agricole durable concours d
	l'innovation agricole durable en
	partenariat avec l'AFD, First Prize
	catégorie Technologie Francophonie
	35.35, Vice-Champion d'Afrique du Pitch
	Hub Africa à Casablanca, Champion
	national Get in the Ring
	2018: Sélectionné pour le programme BM
	de la formation d'entrepreneurs en Corée
	du Sud
	Morty R.		André H.
	Before 2012: TechnoServe programme	2016: TEEP
	laureate, Fonds national de la Jeunesse	2017: Jeune pousse TIC du ministère de
	2011		l'Economie numérique et de la poste
	2015: TEEP		Award
	2016: Business Plan Competition de
	CGECI quarter finalist

A visit to Taveta, Taiti Taveta county, Kenya

Côte d'Ivoire

the Tony Elumelu Foundation (TEF), the French Development Agency (AFD) and the French Institute for Research in Africa, Nigeria (IFRA-Nigeria) signed a partnership agreement that included a knowledge production component. It was decided that this would consist in an academic research work. entrepreneurship programme in four African countries. areas and communities. Sustainable Development Goals of the United Nations (SDGs). -Yaovi KOUASSI

Structure

A rattan workshop in Yaoundé