Anne De Roton

Szilárd Gy

Révész

May

Generalization of the effective Wiener-Ikehara theorem

Keywords: Laplace transform, Dirichlet series, Wiener-Ikehara theorem, Tauberian theorems, slowly decreasing functions, moderately decresing functions, Fourier transform, Lemma of Ganelius, effective error term

lemmas 10 4 Boundedness derived from local assumptions on the Mellin-Stieltjes transform 15 5 Local lemma of "Ganelius-Tenenbaum type" 23 6 Effective Wiener-Ikehara Theorem 26

Introduction

For a carefully written, detailed account of the history and extensive work in the area of the Tauberian theorems of the family usually labeled as Wiener-Ikehara Theorem, we emphasize [3, p. 125]. Our point here is the discussion of general versions of these results.

We first recall Theorem 4.2 of [START_REF] Korevaar | Tauberian Theory -a Century of Developments[END_REF], the integral form of the classical Wiener-Ikehara Tauberian theorem.

Theorem 1.1 (Wiener-Ikehara). Let A(t) vanish on (-∞, 0), be nondecreasing, continuous from the right and such that the Laplace-Stieltjes transform A(s) = ∞ 0 e -st dA(t) is convergent for s > 1. Suppose that for some constant c, the analytic function

G(s) := 1 s + 1 A(s + 1) - c s (1)
has a boundary function in the following sense. For x tending to 1 from the right, the function G x (iy) = G(x + iy) converges to G(1 + iy) either uniformly or in L 1 on every finite interval -T < y < T . Then lim t→+∞ e -t A(t) = c.

Remark 1.1. The convergence of the integral A(s) for s > 1 implies in itself an upper bound for the summatory function A(t) of locally bounded variation on R + (A does not even need to be nondecreasing). Integration by parts indeed leads to the equivalence between the convergence of the Laplace-Stieltjes transform for s > 1 and the estimation |A(t)| ε e (1+ε)t for any positive ε.

Theorem 1.1 has been generalised in different ways.

• The Tauberian condition "A nondecreasing" can be relaxed to some boundedly or slowly decreasing conditions, see [3, p. 135, 141, 142] but in case of bounded decrease, one has to strengthen the regularity of the Laplace-Stieltjes transform on the boundary by assuming its analyticity on the border line.

• One may allow A(s) to have some more general singularity at s = 1 [8, p. 326] and one may take T in Theorem 1.1 to be fixed or even as small as we want but this may lead to a boundedness result rather than a convergence result [3, p. 128, 142].

• There also exists some effective Wiener-Ikehara theorem giving an explicit error term [8, p. 326]. In order to get an explicit error term, we need to know more about the behavior of the Laplace-Stieltjes transform A(s) near and on the border line (T has to be taken large) and we cannot relax too much the Tauberian conditions.

In this work, we give some boundedness theorem for slowly and even boundedly decreasing functions A under some assumption of local behavior of A(s) and an effective Wiener-Ikehara theorem for functions A admitting a slowly or moderately decreasing condition (to be defined later) under some regularity assumptions of A(s) on the border line.

Section 2 will be devoted to the definitions of boundedly, slowly and moderately decreasing functions and to the links between these notions. We recall and state some preliminary results in Section 3. In Section 4, we present the boundedness Tauberian theorem under local assumptions. Finally, in Section 6, we shall present the generalized effective Wiener-Ikehara theorems. The proof of these results will require some lemma close to the Ganelius lemma as generalized by Tenenbaum in [8, p. 328]. This lemma will be given in Section 5.

We shall use in the following the notion of functions of bounded variation, see e.g. [7, pp. 116-]. For a short account of the notion explained in the Lebesgue integral context, i.e. for Lebesgue-Stieltjes integrals, see e.g. Appendix 6 on p. 436-437 of [START_REF] Bingham | Encyclopedia of Mathematics and its Applications 27[END_REF].

Notations. If I is an interval and f a function defined on an interval containing I, we shall write f I for sup x∈I |f (x)|.

We define the Fourier transform of a function f ∈ L 1 (R) as

f (τ ) = +∞ -∞
e -iτ x f (x)dx (τ ∈ R).

Conditions on controlled decrease of functions

Here we recall and introduce some relevant function theoretic notions -that of slowly decreasing functions, introduced by Schmidt in [START_REF] Schmidt | Die Umkehrsätze des Borelschen Summierungsverfahrens[END_REF][START_REF] Schmidt | Über divergente Folgen und lineare Mittelbindungen[END_REF], an essentially weaker version of the property, called here (locally uniformly) bounded decrease and used in a similar form e.g. by Korevaar [3, Proposition 10.2. (i)] and a variation of the notion, which we will term as moderate decrease. Our analysis will cover some aspects not found in the literature and altogether is meant to be a self-contained precursor to the later applications of these properties as Tauberian conditions in Sections 4 and 6.

Let us start with a general analysis of real functions f : [a, ∞) → R, where a > 0 is an arbitrarily fixed parameter. We can always consider, for arbitrary λ ≥ 1, the quantities

ν(f ; λ) := ν(λ) := -inf{f (y) -f (x) : a ≤ x ≤ y ≤ λx} (2)
and

ν(f ; λ) := ν(λ) := -lim inf x→∞ {f (y) -f (x) : a ≤ x ≤ y ≤ λx} (3) = -lim z→∞ inf{f (y) -f (x) : z ≤ x ≤ y ≤ λx} .
The definition of ν(λ) and ν(λ) entails that for any λ ≥ 1 there exists a nonincreasing function Ψ λ with lim x→∞ Ψ λ (x) = 0 satisfying f (y) -f (x) ≥ -ν(λ) -Ψ λ (x) ≥ -ν(λ) whenever a ≤ x ≤ y ≤ λx. (4)

Clearly ν(1) = ν(1) = 0 and 0 ≤ ν(λ) ≤ ν(λ). Both quantities are nondecreasing functions of their variable λ ∈ [1, ∞) and are subadditive in the sense that for any values λ 1 , λ 2 ≥ 1, we have ν(λ 1 λ 2 ) ≤ ν(λ 1 ) + ν(λ 2 ) and similarly for ν. Of course the above remains valid even if the values become infinite, so when ν or ν are functions only in the extended sense. However, once for any value λ 0 > 1 we have ν(λ 0 ) or ν(λ 0 ) finite, then by monotonicity and subadditivity all values remain finite. As we will always consider locally bounded functions f , in fact for us finiteness of ν and ν will be equivalent, although in general there exist functions which are unbounded and oscillate badly at some point and thus have ν(λ) < +∞ whereas ν(λ) = +∞.

The first notion which we will use further as a condition on the decrease of a function is thus expressed by the finiteness of these characteristics. Definition 2.1. We say that a real function f defined on [a, +∞) is boundedly decreasing if we have ν(f ; λ) < ∞ for some -and hence for allλ > 1.

Applying Definition 2.1 on intervals of the form [e k x, e k+1 x] with λ = e, we infer for any λ ≥ e the lower bound

f (y) -f (x) ≥ -ν(f ; e) log λ whenever (a ≤)x ≤ y ≤ λx. (5)
In this work we basically consider functions F with F (x)/x boundedly decreasing on [a, ∞) for some a > 0. Taking x = a in inequality (5), we see that if F (x)/x is a boundedly decreasing function, then for y ≥ a

F (y) ≥ -M y(1 + log y -log a) ≥ -2M y max(1 -log a, log y) (6)
with M := M (F, a) := ν(F (x)/x; e) + F (a) -/a. Subadditivity, nonnegativity and monotonicity entails that whenever ν is continuous at 1, that is when we have lim λ→1 ν(λ) = ν(1) = 0, then ν is continuous everywhere on [1, ∞). (It is clear that this limit must be at least 0, the question is if it is indeed zero.) The situation is the same for ν. Definition 2.2 (Schmidt). Let a > 0. We say that a real function f defined on [a, +∞) is slowly decreasing if we have

lim λ↓1 lim inf x→∞ inf x≤y≤λx {f (y) -f (x)} = 0, (7)
that is if lim λ↓1 ν(λ) = 0, and consequently ν is a nonnegative, finite, continuous, subadditive, nondecreasing function of λ ∈ [1, ∞).

Ever since R. Schmidt's pioneering work [START_REF] Schmidt | Die Umkehrsätze des Borelschen Summierungsverfahrens[END_REF][START_REF] Schmidt | Über divergente Folgen und lineare Mittelbindungen[END_REF], slowly decreasing functions have been used extensively in the context of Tauberian theorems, see e.g. [10, Ch. III, §8], [1, p. 41], [3, p. 33, p. 69]. Properties of slowly decreasing functions were already well-explored by Schmidt [START_REF] Schmidt | Die Umkehrsätze des Borelschen Summierungsverfahrens[END_REF], [START_REF] Schmidt | Über divergente Folgen und lineare Mittelbindungen[END_REF], and can be found in several places, see e.g. [10, p. 298].

In the following we will encounter various restrictions on the decrease of a function as Tauberian conditions in deriving an effective Wiener-Ikehara Theorem. A usual condition in this context, (and one used also in Karamata's Theorem) is that F (x)/(x log m x) is slowly decreasing. One would think at first sight, that once F (x)/x is slowly decreasing, so does the further divided F (x)/(x log m x), but caution is aroused in seeing that division by a nice increasing function in itself does not preserve slow decrease, as is the case with division by x, see Corollary 2.1.

The next proposition will give, for some nondecreasing function , the order estimates on a general slowly or boundedly decreasing function q(x) that guarantee that also q/ be slowly or boundedly decreasing. Proposition 2.1. Let a > 0 be a real number and : [a, +∞) → (0, ∞) be a positive, nondecreasing, differentiable function. If / is nonincreasing, then for all slowly (respectively boundedly) decreasing functions q(x) satisfying q(x) = O( 2 (x)/(x (x))) we have q/ also slowly (respectively boundedly) decreasing on [a, +∞).

Remark 2.1. Analogous results can be derived for more general functions . Since all the functions which we shall use satisfy the assumptions in the proposition, we do not state the proposition in its full generality.

Corollary 2.1. Let q be a slowly (respectively boundedly) decreasing function on [1, +∞). If m is a positive real number such that q(x) = O(log m+1 x) for x ≥ e, then also q(x)/ log m x is slowly (respectively boundedly) decreasing on [1, +∞) whereas if q(x) = O(x) for x ≥ 1, then also q(x)/x is slowly (respectively boundedly) decreasing on [1, +∞).

Then again, for any positive function ω tending to infinity, however slowly, there exists a slowly decreasing function q on [1, +∞) such that q(x) = O(xω(x)) for x large enough, but q(x)/x is not even boundedly decreasing.

Note that the first statement of the Corollary shows that for certain functions , q(x) = O( (x)) is not the ultimate (most general) order condition on a slowly decreasing function q to ensure slow decrease of q/ ; but the last statement exemplifies that for (x) = x it is.

Proof of the Corollary. Proposition 2.1 with the choices (x) = log m x and (x) = x, respectively, yields the first two assertions.

To see the last, we first define a nondecreasing function s 0 which tends to infinity as x tends to infinity by s 0 (x) := inf x ≥x ω(x ). Then we define the function s on[1, ∞) by s(1) = 0 and for any integer n ≥ 0, s(x) = min(s 0 (x), s(e n ) + 1),

x ∈ (e n , e n+1 ] .

The function s is nondecreasing and unbounded. Furthermore it satisfies 0 ≤ s ≤ s 0 ≤ ω, and s(ex) ≤ s(x) + 2 for x ≥ 1. The function q defined by q(x) := e log x s(x) satisfies q(x) = O(xω(x)) and is nondecreasing. Nevertheless, q(x)/x is not slowly decreasing since for arbitrary 1 < λ < e,

inf n∈N q(λe n ) λe n - q(e n ) e n = inf n e n s(λe n ) λe n - e n s(e n ) e n ≤ inf n 2 + (1 -λ)s(e n ) λ = -∞. Proof of Proposition 2.1. Let C be a constant satisfying q(x) ≤ C 2 (x) x (x) for any x ≥ a.
We write for arbitrary a ≤ x ≤ y ≤ λx, λ > 1

q(y) (y) - q(x) (x) = q(y) -q(x) (y) -q(x) 1 (x) - 1 (y) . ( 8
)
By the Mean Value Theorem, there exists ξ ∈ (x, y) such that

q(x) 1 (x) - 1 (y) = |q(x)| (x) (y) -(x) (y) = |q(x)| (x) (ξ)(y -x) (y) ≤ |q(x)| (x) (ξ) (ξ) (λ -1)x ≤ (λ -1) |q(x)|x (x) 2 (x) ≤ C(λ -1),
in view of the monotonicity of , / and the growth order assumption on q. Writing this last formula in (8) yields

q(y) (y) - q(x) (x) ≥ - {q(y) -q(x)} - (a) -C(λ -1).
This gives q(y)

(y) - q(x) (x) ≥ - ν(λ) (a) -C(λ -1)
in case q is boundedly decreasing and

lim inf x→+∞ inf x≤y≤λx q(y) (y) - q(x) (x) ≥ - 1 (a) ν(λ) -C(λ -1),
which tends to 0 when λ → 1 in case q is slowly decreasing . That concludes the proof.

We introduce now a new class, that of moderately decreasing functions, which will have a natural occurrence in our analysis. Some similar, but less general, condition for functions were considered by Korevaar in [START_REF] Korevaar | Tauberian Theory -a Century of Developments[END_REF], p.382. Definition 2.3. Let a ≥ 0 be arbitrary. A real function F on [a, +∞) is moderately decreasing if there exist some positive constants B 1 , B 2 such that for any pair (u, v) of real numbers satisfying v ≥ 0 and u ≥ a we have

F (u + v) -F (u) ≥ -B 1 v -B 2 max(1, u)ϕ(u) ( 9
)
where ϕ is a nonincreasing function on [a, +∞) satisfying ϕ(x) = 1 for any x ∈ [a, max(1, a)] and lim u→∞ ϕ(u) = 0.

From the definition applied to u = a and v = x -a ≥ 0 one obtains immediately that for x ≥ max(a, 1)

F (x) ≥ -B 1 (x -a) -B 2 max(a, 1) -F (a) -≥ -Bx (10)
with

B := B(F, a) := B(F, a, B 1 , B 2 ) := B 1 + F (a) -+ B 2 .
Note that upper bounds for F (x) cannot be derived from the moderately decreasing property, since any increasing function is necessarily moderately decreasing.

In Theorem 4.1 we will show, however, that in case we have some control on the Mellin-Stieltjes transform of a moderately decreasing function A, then it entails some respective non-trivial upper bound on A.

Remark 2.2. As proved in the proof of Corollary 2.1, there are some increasing (hence moderately decreasing) functions F such that F (x)/x is not even boundedly decreasing, hence not slowly decreasing either.

Conversely, there are some functions F which are not moderately decreasing whereas F (x)/x is slowly decreasing. Even more so, F (x)/x can be chosen very slowly decreasing as introduced e.g. in [3, formula (10.2), p. 143] and meaning that ν(λ) = 0 for some -and hence (in view of monotonicity and subadditivity) for allλ > 1.

For instance if F (x) = -x log x/ log log x, then F (x)/x is very slowly decreasing, but for u large and v = u/ √ log u, condition (9) fails for F , moreover, not even condition (10) holds true for x large.

Nevertheless, for well-bounded functions there is some connection between these notions of controlled decrease. Proposition 2.2. Let a and A be real numbers satisfying 1 ≤ a ≤ A and A ≥ e, m > 0 and (x) := log m x. If a real function F defined and moderately decreasing on [a, ∞) satisfies F (x) = O(x (x)) for x ≥ A, then F (x)/(x (x)) is slowly decreasing on [A, ∞). However, the converse fails to hold, as there exist functions F (x) = O(x) with F (x)/x slowly decreasing, but F (x) not moderately decreasing.

Remark 2.3. Corollary 2.1 gives that if F (x) = O(x) with F (x)/x slowly decreasing, then F (x)/(x (x)) is also slowly decreasing. Therefore the second assertion really proves that the converse of the first one does not hold.

Proof of Proposition 2.2. Assume that F satisfies (9) and |F (x)| ≤ Cx (x) for x ≥ A and some positive constant C. Consider for any 1 < λ < log 2 and A ≤ x ≤ y ≤ λx the estimation To disprove a converse implication, first we define a slowly decreasing and bounded function f (x) with ν(λ) to be large in a sense to be made more precise later.

F (y) y (y) - F (x) x (x) = F (y) -F (x) y (y) - y (y) -x (x) y (y) F (x) x (x) ≥ - B 1 (y -x) + B 2 ϕ(x)x y (y) -C y -x y + x y 1 - (x) (y) ≥ -B 2 ϕ(x) -(B 1 + C) λ -1 λ -C 1 - (x) (λx) ,
Let (m n ) be a sequence of integers at least 2 covering all the integers of [2, +∞) infinitely often. We define the function f on [1, +∞) as follows:

f (x) = - √ mn 2 n (x -2 n ) if x ∈ I n := [2 n , mn+1 mn 2 n ) -1 √ mn if x ∈ J n := [ mn+1 mn 2 n , 2 n+1 )
.

Clearly -1 √ 2 ≤ -1 √ mn ≤ f ≤ 0 on any interval [2 n , 2 n+1 ), so f (x) = O(1)
. Next we show that f is indeed slowly decreasing. For any x ∈ [1, +∞) and any λ ∈ (1, 4/3], the interval [x, λx] can mesh with at most one interval I n , in which case we have for any y ∈ [x, λx],

f (y) -f (x) ≥ - √ m n 2 n min (y -x, |I n |) ≥ - √ m n 2 n min (λ -1)2 n+1 , 2 n m n ≥ -min 2(λ -1) √ m n , 1 √ m n ≥ -2(λ -1)
since 2 n-1 < x < 2 n+1 . In case [x, λx] does not mesh with any interval I n , this is also satisfied since f (y) -f (x) = 0 for any y ∈ [x, λx]. This finally proves that ν(λ) ≤ 2(λ -1) and thus f is slowly decreasing. So now we define F (x) = xf (x). Then F (x) = O(x), F (x)/x is slowly decreasing, and it remains to see that nevertheless F (x) is not moderately decreasing. To do this, we show that for any given fixed positive constant C, there exists some c > 0 such that the difference F (x + y) -F (x) + Cy stays below -cx for some choice of x, y tending to ∞.

Making use of the fact that F (x) ≤ 0, we write

F (x + y) -F (x) ≤ (x + y) F (x + y) x + y - F (x) x = (x + y) (f (x + y) -f (x)) .
Let us fix C > 0 and choose N ∈ N so that N > 4C 2 and let n k = n k (N ) be a strictly increasing sequence with m n k = N (k = 1, 2, . . . ). Now with

x k = 2 n k and y k = 1 N 2 n k = |I n k |, we have f (x k + y k ) -f (x k ) = -1/ √ N , thus F (x k + y k ) -F (x k ) + Cy k ≤ - x k + y k √ N + C x k N ≤ - x k √ N + √ N 2 x k N = - x k 2 √ N , because C < √ N /2. Therefore, choosing c := 1/(2 √ N ) yields the assertion.
Remark 2.4. With Corollary 2.1, we proved that under the assumption F (x) = O(x log m x), the assumption F (x)/x slowly (respectively boundedly) decreasing implies that F (x)/(x log m-1 x) is slowly (respectively boundedly) decreasing.

Preliminary lemmas

We recall here a few classical tools in analysis and formulate some technical lemmas.

We define, for T > 0 and u ∈ R, the usual Fejér kernels as

χ(u) := 1 2π sin(u/2) u/2 2 and χ T (u) := T χ(T u). (11)
These kernels satisfy the following properties:

R χ T (u)du = 1, (12)
χ T (τ ) = χ(τ /T ) = (1 -|τ |/T ) + , (13)
and for q > 0,

I q := |u|>q/T χ T (u)du = |u|>q χ(u)du ≤ 4 πq . (14)
We now state a lemma which will come in handy in Sections 4 and 6.

Lemma 3.1. Let T > 0 be a positive constant and f : R → R be a bounded function of locally bounded variation. Let λ 1 > 0, λ 2 ≥ 0 and λ 3 ≥ 0 be some constants and assume that f satisfies

f (u + v) ≥ λ 1 f (u) as soon as f (u) > λ 2 and v ∈ [0, 10/T ]. (15)
Assume furthermore that

• either f (u) ≥ -λ 3 for any u ∈ R (Case 1), • or λ 1 > 4 5π -4
and the function g defined on R by g(u) = -f (-u)

satisfies (15) (Case 2).
Then we have

f ∞ ≤ k 1 T -T f (τ ) dτ + k 2 with k 1 := 5 2(5π -4)λ 1 , and
k 2 := max λ 2 , 4λ 3 (5π -4)λ 1 , λ 3 .
in the first case and

k 1 := 5 2(5π -4)λ 1 -8 , and
k 2 := λ 2 .
in the second one.

Proof. In both cases, we may assume that f ∞ = sup u f (u). Otherwise the lower bound f (u) ≥ -λ 3 would give the result in the first case (as

λ 3 ≤ k 2 )
and in the second case, we could consider g defined by

g(u) = -f (-u) instead of f . Furthermore, if f ∞ ≤ λ 2 , we are done as λ 2 ≤ k 2 in both cases.
So for the rest of the proof we assume, as we may, that f ∞ = sup u f (u) > λ 2 . For u ∈ R, we estimate the integral

J(u) := R χ T (v)f (u + 5/T + v)dv,
where T > 0 and χ T is defined in [START_REF] Whittaker | A course of modern analysis. Fourth[END_REF]. Plancherel's formula and (13) lead to the upper bound

|J(u)| = 1 2π ∞ -∞ χ T (τ ) f (τ )e i(5/T +u)τ dτ ≤ 1 2π T -T f (τ ) dτ. (16)
The integral J(u) is real, and cutting the integral to parts over [-5/T, 5/T ] and outside that, we find the lower estimation

J(u) ≥ (1 -I 5 ) min v∈[0,10/T ] f (u + v) + I 5 min t ∈[u,u+10/T ] f (t), (17)
where

I 5 is defined in (14). Now for any 0 < ε < 1 -λ 2 / f ∞ we take some u ∈ R with (λ 2 <) (1 -ε) f ∞ < f (u) ≤ f ∞ . For such a u, we have f (u + v) ≥ λ 1 f (u) for all v ∈ [0, 10/T ] by assumption, thus (17) gives J(u) ≥ (1 -I 5 )λ 1 f (u) + I 5 min R f (18)
In Case 1, we use min R f ≥ -λ 3 whereas in Case 2 we use min

R f ≥ -f ∞ > -f (u)/(1 -ε).
Combining this with ( 18) and ( 16), we are led to

f (u) ≤ λ 4 T -T f (τ ) dτ + λ 5 with λ 4 = 1 2π ((1 -I 5 )λ 1 ) -1 ≤ 5 (2(5π -4)λ 1 ) -1 , λ 5 = λ 3 I 5 ((1 -I 5 )λ 1 ) -1 ≤ 4(5π -4) -1 (λ 3 /λ 1 )
in Case 1 and in Case 2

λ 4 = 1 2π ((1 -I 5 )λ 1 -I 5 /(1 -ε)) -1 ≤ 5 (2(5π -4)λ 1 -8/(1 -ε)) -1 , λ 5 = 0
where ε is taken small enough so that (5π

-4)λ 1 (1 -ε) > 4. (Recall that in Case 2, λ 1 > 4/(5π -4) by condition).
Taking into account also the already settled case when f ∞ < λ 2 , and letting ε tend to 0, we therefore get the announced inequalities. Now we introduce some integrals which we shall need in Sections 4 and 6. Let m, T be positive real numbers and σ be a nonnegative real number. We consider the following quantities.

W m (σ, T ) := T 0 dt |σ + it| m , Z m (σ, T ) := T 0 dt |σ + it| m |1 + σ + it| . (19)
The integrals W m (σ, T ) satisfy the following properties:

W m (σ, T ) = σ 1-m W m (1, T /σ); (20)
W m (σ, T ) m      σ 1-m if m > 1 log 1 + T σ if m = 1 T {max(T, σ)} -m if m < 1 . ( 21
)
When m > 1, we also have

W m (1, T ) < W m (1, ∞) < m m -1 . (22)
Analogously, a standard calculation furnishes for 0 < σ < 1 (with standing for and together)

Z m (σ, 1) m Z m (σ, ∞) m      σ 1-m if m > 1 log(1 + 1/σ) if m = 1 1 if 0 < m < 1 (23)
and

Z 0 (σ, T ) log(1 + T ). (24)
We finally state a lemma providing some upper bounds for a certain function β and its differences. Lemma 3.2. For any ω > -1 and t ∈ R, let β(ω, t) be defined by

β(ω, t) := t ω Γ(ω+1) e -t (1 -e -t ) (t > 0) 0 (t ≤ 0). ( 25
)
Then for ω > -1, x ∈ R and y > 0, we have

|β(ω, x)| ≤ 1/ √ π, |β(ω, x + y) -β(ω, x)| ≤ y ω+1 + 2y/ √ π. (26)
Proof. In all of this argument we will repeatedly use that for any ξ > 0

max t>0 t ξ e -t = ξ ξ e -ξ , (27)
and (see [11, §12.33])

Γ(ξ) > ξ ξ-1/2 e -ξ √ 2π. ( 28
)
We also recall the functional equation of Γ: for any ξ ∈ Z -, Γ(ξ+1) = ξΓ(ξ).

Let ω > -1 and t be real numbers. First we give an upper bound for |β(ω, t)|.

If t ≤ 0, then β(ω, t) = 0 so we will restrict to t > 0, where β(ω, t) > 0 and it suffices to estimate β(ω, t) from above.

• If -1 < ω ≤ 1, we use 1 -e -t < t and ( 27) and (28) with ξ = ω + 1.

We obtain

β(ω, t) < t ω+1 e -t Γ(ω + 1) ≤ (ω + 1) ω+1 e -ω-1 (ω + 1) ω+1/2 e -ω-1 √ 2π = √ ω + 1 √ 2π ≤ 1 √ π .
• If ω > 1, we use 1 -e -t < 1 and ( 27) and (28) with ξ = ω. We obtain

β(ω, t) < t ω e -t Γ(ω + 1) ≤ ω ω e -ω ωΓ(ω) ≤ ω ω e -ω ω ω+1/2 e -ω √ 2π = 1 √ 2πω < 1 √ 2π .
Now consider the change of the function, hence the derivative of β for t > 0:

β (ω, t) = [2te -t -t + ω(1 -e -t )]t ω-1 e -t Γ(ω + 1
) .

• For -1 < ω ≤ 0, we use t(e -t -1) ≤ 0 first and then 1 -e -t > te -t and e -t < 1 to infer

β (ω, t) < [te -t + ω(1 -e -t )]t ω-1 e -t Γ(ω + 1) < (1 + ω)t ω Γ(ω + 1) ≤ (1 + ω)t ω ,
as Γ(ξ) ≥ 1 when 0 < ξ < 1. For x ∈ R and y > 0 this yields

β(ω, x+y)-β(ω, x) ≤ 1 + ω max(0,x+y) max(0,x) t ω dt < 1 + ω y 0 t ω dt ≤ y ω+1 .
Next, for a lower estimation using te -t > 0 and 1 -e -t < t we obtain

β (ω, t) > -t(1 -e -t ) + ωt Γ(ω + 1) t ω-1 e -t > (-t + ω)t ω e -t Γ(ω + 1) .
As before, integration yields for x ∈ R and y > 0

β(ω, x + y) -β(ω, x) ≥ 1 Γ(ω + 1) ω ω + 1 y ω+1 - ω + 1 e ω+1 y
where we used (27) to estimate the last integral. For the first coefficient, using that Γ(ξ) is already increasing for ξ ≥ 2, we get

ω (ω + 1)Γ(ω + 1) = ω Γ(ω + 2) = ω(ω + 2) Γ(ω + 3) > -1 Γ(ω + 3) > -1 Γ(2) = -1.
For the second coefficient, we apply (28) with ξ = ω + 1 and get

1 Γ(ω + 1) ω + 1 e ω+1 ≤ 1 + ω 2π ≤ 1 √ 2π . Whence for -1 < ω ≤ 0 we get |β(ω, x+y)-β(ω, x)| ≤ y ω+1 +y/ √ 2π.
• Consider now the case ω ≥ 2. We use (e -t -1) < 0 and 1 -e -t < 1 and then we use ( 27) and (28). We get

β (ω, t) < [te -t + ω]t ω-1 e -t Γ(ω + 1) < t ω e -2t ωΓ(ω) + t ω-1 e -t (ω -1)Γ(ω -1) ≤ 2 -ω 1 √ 2πω + 1 2π(ω -1) ≤ 2 -2 2 + 1 √ 2 1 √ π ≤ 1 √ π .
This entails β(ω, x + y) -β(ω, x) < y/ √ π.

From the other direction, we have

β (ω, t) ≥ - e -t t ω ωΓ(ω) ≥ - 1 √ 2πω ≥ - 1 2 √ π thus β(ω, x + y) -β(ω, x) ≥ -y/(2 √ π). Whence for ω ≥ 2 we get |β(ω, x + y) -β(ω, x)| ≤ y/ √ π.
• Finally, let us assume that 0 < ω < 2. Similarly as before, we use ( 27) and (28) to get for t > 0

β (ω, t) ≤ [te -t + ωt]t ω-1 e -t Γ(ω + 1) = t ω e -2t Γ(ω + 1) + t ω e -t Γ(ω) ≤ t ω Γ(ω + 1) + ω 2π ≤ (ω + 1)t ω Γ(ω + 2) + 1 √ π ≤ (ω + 1)t ω + 1 √ π .
This entails β(ω, x + y) -β(ω, x) ≤ y ω+1 + y/ √ π. We also have

β (ω, t) ≥ - t ω+1 e -t Γ(ω + 1) ≥ - ω + 1 2π > - 2 √ π yielding β(ω, x + y) -β(ω, x) ≥ -2y/ √ π.

Boundedness derived from local assumptions on the Mellin-Stieltjes transform

We can find in the book of Korevaar, see [ Assume further that the Mellin-Stieltjes transform A(s) := ∞ 1 x -s dA(x) converges in the half plane s > 1 and satisfy for all 0 < σ < 1

T 0 |A(1 + σ + it)| |1 + σ + it| dt ≤ K 1 Z m (σ, ∞) if m > 0, Z 0 (σ, T ) if m = 0 (29)
where Z m is defined in (19) and K 1 is some positive constant. Then we have A(x) = O(x log (m-1) + x) for all x ≥ e if m = 1 and A(x) = O(x log log x) for all x ≥ e e if m = 1, where the implicit constant of the O symbol depends explicitly on m, on K 1 and • on A(x)/x 1≤x≤e 1+50/T and on ν(A(x)/x; e 10/T ) under Condition 1,

• on A(x)/x 1≤x≤e 1+c/T with c = 10 max(5, 1 + (m -2) + / log 2) and on ν(A(x)/(x log (m-2) + x); e 10/T ) under Condition 2,

• on A(x)/x 1≤x≤e , on T (more precisely, on e 10/T ) and on the constants B 1 , B 2 under Condition 3.

In case m = 0, we also have a dependence on T with both the boundedly and the moderately decreasing conditions.

Remark 4.1. In retrospect we can observe that under the condition of (29) the first assumption, i.e. that "A(x)/x is boundedly decreasing", is stronger (more restrictive) than the second one stating "A(x)/(x log (m-2) + x) is boundedly decreasing". Indeed, if the first assumption is satisfied, then the above theorem furnishes the O-result which in turn entails this second assumption as noticed in Remark 2.4.

Proof. Concerning the bounded decrease condition, the case m = 0 is given in the book of Korevaar, see [3, Proposition III.10.2, (i) p. 143]. We will generalize his arguments in this proof. For σ > 0, we define the function h σ on R by h σ (t) := e -(1+σ)t A(e t ). According to Remark 1.1, the function h σ is bounded. To prove the theorem, it is enough to prove with µ := (m -1) + and some σ 0 ∈ (0, 1) that it holds

h σ ∞ σ -µ if m = 1 log(1 + 1/σ) if m = 1 (0 < σ < σ 0 ). (30)
Indeed, with t := log x, we have A(x) = h 1/t (t)e t+1 , thus (30) implies for

x > e 1/σ 0 ,

A(x) x log µ x if m = 1 x log log x in case m = 1 ,
while for x ≤ e 1/σ 0 , local boundedness of A ensures the statement provided that we allow a dependence of the implied constant on A(x)/x [1,e 1/σ 0 ] . For a parameter λ ≥ 0, to be fixed later, we define the sets S ± µ (λ) as

S + µ (λ) := {t ≥ 0 : A(e t ) > λe t t µ }, S - µ (λ) := {t ≥ 0 : A(e t ) < -λe t t µ }.
If t ≥ 0 and t ∈ S µ (λ) := S + µ (λ) ∪ S - µ (λ), then with the convention 0 0 = 1

|h σ (t)| ≤ λt µ e -σt ≤ λ σ µ e -v v µ R + = λ σ µ µ e µ µ,λ σ -µ .
Now we put µ 0 = (µ -1) + = (m -2) + under Condition 2 and µ 0 = 0 under Conditions 1 and 3.

• Under Condition 1 and 2, we put

λ 2 := max A(x) x [1,exp(1+ 10 T max{5, µ 0 log 2 +1})] , λ µ e µ σ -µ . ( 31
)
Let u be a real number such that |h σ (u)| > λ 2 .

In view of |h σ (u)| < |A(e u )|/e u , we necessarily have u ≥ 1 + 50/T . Moreover, if θ is the sign of h σ (u), then u ∈ S θ µ (λ) and, as obviously

S θ µ (λ) ∩ [1, +∞) ⊂ S θ µ (λ)
for any µ ≤ µ, we have as well u ∈ S θ µ 0 (λ). By condition of bounded decrease we have for e ≤ x ≤ y ≤ e 10/T x A(y) y(log y) µ 0 -

A(x) x(log x) µ 0 ≥ -ν(e 10/T ), (32)
where ν(e 10/T ) := ν(A(x)/(x log µ 0 x); e 10/T ).

Let v be a real number satisfying 0 ≤ v ≤ 10/T . We apply (32) with (x, y) = (e u , e u+v ) if h σ (u) > 0 or (x, y) = (e u-v , e u ) if h σ (u) < 0.

Note that u ≥ 1 + 50/T > 1 and u -v ≥ 1 + 40/T > 1, so in case λ > 0, we infer for any u and v as above

θh σ (u + θv) = θ (u + θv) µ 0 e σ(u+θv)
A(e u+θv ) e u+θv (u + θv) µ 0 -A(e u ) e u u µ 0 + A(e u ) e u u µ 0 ≥ θe -σ(u+θv) -θν(e 10/T ) +

A(e u ) e u u µ 0 (u + θv)

µ 0 ≥ θe -σ(u+θv) 1 - ν(e 10/T ) λ A(e u )(u + θv) µ 0 e u u µ 0 ≥ e -θσv 1 - ν(e 10/T ) λ |h σ (u)| u + θv u µ 0 .
Now we chose the parameters λ and σ 0 as λ := 5ν(e 10/T ) and σ 0 := min(1, log 2 10 T ). When θ = +, we obtain from the above for any 0 < σ < σ 0 the estimate

h σ (u + v) ≥ 4 5 e -σ10/T h σ (u) ≥ 2 5 h σ (u). (33)
Similarly, for θ = -, using u > (1+µ 0 / log 2)10/T we obtain for σ > 0

-h σ (u -v) ≥ e σv 4 5 |h σ (u)| u -10/T u µ 0 ≥ 2 5 (-h σ (u)) , (34)
because 1 + w ≤ e w implies q+1 q ≤ exp( 1 q ) and q q+1 ≥ exp( -1 q ) which yields

u -10/T u µ 0 ≥ µ 0 / log 2 µ 0 / log 2 + 1 µ 0 ≥ exp -µ 0 µ 0 / log 2 = 1 2 .
Inequalities (33) and (34) remain true when ν(e 10/T ) = 0 and hence λ = 0.

We have just proved that if 0 < σ < σ 0 = min(1, log 2 10 T ), then for any u such that θh σ (u) > λ 2 and any v ∈ [0, 10/T ], we have θh σ (u + θv) ≥ 2 5 θh σ (u). So choosing λ 1 := 2/5 and noting that 2/5 > 4/(5π -4), an application of the second part of Lemma 3.1 to h σ leads to

h σ ∞ T -T h σ (τ ) dτ + λ 2 (0 < σ < σ 0 ), (35)
with an absolute implicit constant.

• Assume now that Condition 3 holds. We take λ > 0.

Let u be a real number such that h σ (u) > λ, then u ∈ S + 0 (λ). By condition of moderate decrease, we have for any v ≥ 0 A(e u+v ) -A(e u ) ≥ -B 1 (e v -1)e u -B 2 ϕ(e u )e u .

If v ∈ [0, 10/T ] and λ > 0, we can infer

h σ (u + v) = e -σ(u+v) A(e u+v ) -A(e u ) e u+v + e -v A(e u ) e u ≥ e -σ(u+v) -B 1 (1 -e -v ) -B 2 e -v ϕ(e u ) + e -v A(e u ) e u ≥ e -σ(u+v)-v -max(B 1 , B 2 )e 10/T + A(e u ) e u ≥ e -(σ+1)v 1 - max(B 1 , B 2 )e 10/T λ h σ (u).
Choosing λ := 5 max(B 1 , B 2 )e 10/T , we get for 0 < σ < 1 and u and v as above

h σ (u + v) ≥ 4 5 e -(σ+1)10/T h σ (u) ≥ 4 5 e -20/T h σ (u). (36)
This last inequality remains valid when λ = 0, too. Now we use [START_REF] Wiener | Tauberian theorems[END_REF] to see that h σ (u) > -B with B := B(A, 1). Finally, we apply the first part of Lemma 3.1 with λ 2 = λ,λ 1 = 4 5 e -20/T and λ 3 = B. Thus we get for 0 < σ < 1,

h σ ∞ T -T h σ (τ ) dτ + λ 2
where the implicit constant depends on e 20/T and B.

Since in all cases λ 2 σ -µ , it remains to estimate the integral

T -T h σ (τ ) dτ . As h σ (τ ) = A(1 + σ + iτ ) (1 + σ + iτ ) , (37)
using the hypothesis (29) on A, we get

T -T h σ (τ ) dτ = T -T A(1 + σ + iτ ) 1 + σ + iτ dτ Z m (σ, ∞) if m > 0, Z 0 (σ, T ) if m = 0.
Applying (23) and ( 24) gives (30), whence the result.

We shall now discuss the condition (29) on A. First, we give some more natural but stronger condition under which the conclusion holds. Proof. Developing A in a neighborhood of s = 1 gives for some T > 0,

|A(s)| 1 |s -1| m (1 < (s) < 2, | (s)| ≤ T ). (38)
Clearly, the condition (38) entails (29), hence the result follows except for m = 1.

In case of a simple pole with residue r, however, we can apply this result to A(x) -rx with m = 0. To this note that together with A(x)/x also (A(x) -rx)/x is boundedly decreasing, and with A(x) also A(x) -rx is moderately decreasing, while the Mellin-Stieltjes transform of A(x) -rx is A(s) -r/s, that is regular. So under either assumptions on the controlled decrease of A, we get

A(x) -rx = O(x), thus A(x) = O(x).
In the effective Wiener-Ikehara Theorem (Theorem 6.1), the condition to get an estimate for A(x) is

η(σ, T ) := T 0 A(1 + σ + it) 1 + σ + it - A(1 + 2σ + it) 1 + 2σ + it dt σ -(m-1) + (39)
for some T > 0 and for any 0 < σ < 1. For concrete Laplace transforms the expression in Condition (29) is larger (hence the condition is stronger), than η(σ, T ) because we usually find considerable cancelation in the latter. By the triangle inequality, if (29) holds, then we also have

η(σ, T ) Z m (σ, ∞) + Z m (2σ, ∞) Z m (σ, ∞) if m > 0 Z m (σ, T ) + Z m (2σ, T ) Z m (σ, T ) if m = 0 .
The following lemma states that the converse is also true. x -s dA(x) converges in the half plane s > 1. Assume further that for some fixed nonnegative value of T , there exist some positive constants µ and k such that η(σ, T ) ≤ kσ -µ for 0 < σ < 1.

Then there exists a positive constant K such that for all σ ∈ (0, 1) we have the estimate

T 0 A(1 + σ + it) 1 + σ + it dt ≤ K Z µ+1 (σ, ∞). ( 40
)
If µ > 0, we have K = kk with k depending explicitly only on µ.

In case µ = 0 we can make the constant K explicit if either Condition 1 or 2 for some m > 0 or Condition 3 of controlled decrease in Theorem 4.1 holds. In this case the constant K depends explicitly on the parameters k, T and A(2) and on the constant M (A, 1) under Condition 1, or on M (A/ log (m-2) + , e), m and A [1,e] under Condition 2, or on B(A, 1) under Condition 3.

Proof. First let µ > 0. By a repeated application of the triangle inequality and using (37) we obtain for arbitrary N ∈ N

T 0 A(1 + σ + it) 1 + σ + it dt ≤ N -1 n=0 η(2 n σ, T ) + T 0 A(1 + 2 N σ + it) 1 + 2 N σ + it dt ≤ N -1 n=0 k(2 n σ) -µ + T 0 | h 2 N σ (t)|dt ≤ k 1 -2 -µ σ -µ + T h 2 N σ ∞ .
But for any θ > 0 we have

h θ ∞ ≤ h θ 1 = ∞ 0 |A(e u )|e (-1-θ)u du ∞ 0
e (1-θ)u du using Remark 1.1 with ε = 1, so according to the monotone convergence theorem (or the Lebesgue convergence theorem) lim θ→∞ h θ 1 = 0 and we are led to

T 0 A(1 + σ + it) 1 + σ + it dt ≤ k 1 -2 -µ σ -µ ,
which gives the assertion for µ > 0 in view of (23).

In case µ = 0 we start our calculation similarly, but fix the value of N as N := log 2 (1/σ) and thus have 2 N σ ≥ 1, always. The above calculation then leads to

T 0 A(1 + σ + it) 1 + σ + it dt ≤ kN + T ∞ 0 |A(e u )|e (-1-2 N σ)u du ≤ k 1 + log 2 1 σ + T ∞ 0
|A(e u )|e -2u du.

Clearly, the last integral is bounded in virtue of Remark 1.1, so at least with some ineffective constant K the assertion in (40) follows. However, to make the constant explicit, we need to invoke the above explicit calculations with functions of controlled decrease. Let us denote µ 0 = 0 under Conditions 1 and 3 and µ 0 = (m-2) + under Condition 2 and

k * :=    M (A, 1)
under Condition 1 M (A/ log (m-2) + , e) + A [1,e] under Condition 2 B(A, 1)

under Condition 3.

By ( 6) and ( 10), we always have A(x) > -k * x(1 + log x) 1+µ 0 for x ≥ 1 and so in view of |A(e u )| = A(e u ) + + A(e u ) -= A(e u ) + 2A(e u ) -we are led to

∞ 0 |A(e u )|e -2u du ≤ ∞ 0 A(e u )e -2u du + 2k * ∞ 0 (u + 1) 1+µ 0 e -u du ≤ ∞ 0 A(e u )e -2u du + 2k * eΓ(2 + µ 0 ) = A(2) 2 + 2eΓ(2 + µ 0 )k * =: K * ,
just denoting this last constant as K * . So now we have for 0 < σ < 1

T 0 A(1 + σ + it) 1 + σ + it dt ≤ k 1 + log 2 1 σ + K * T ≤ 2 log 2 (K * T + k) log 1 + 1 σ .
Taking into account Z 1 (σ, ∞) log(1 + 1/σ) from ( 23) we get

T 0 A(1 + σ + it) 1 + σ + it dt ≤ K * * log 1 + 1 σ ≤ K Z 1 (σ, ∞),
with K * * and hence K depending explicitly on the parameters given.

(σ, T ) ≤ K 2 σ -(m-1) + (0 < σ < 1) (41)
for some positive constant K 2 . Then we have

|A(x)| ≤ K 3 x log m-1 x if m > 1 K 3 x log log x if 0 < m ≤ 1 (x ≥ e e ) . (42)
Here the constant K 3 is explicit and depends on |A(2)| (if m ≤ 1) and on the same parameters as in Theorem 4.1 but now dependence on K 1 in (29) is replaced by dependence on K 2 in (41).

Proof. We apply Lemma 4.1 with µ = (m -1) + , and thus obtain

T 0 A(1 + σ + it) 1 + σ + it dt Z µ+1 (σ, ∞). ( 43
)
An application of Theorem 4.1 then gives the result. Note that in case µ = 0, Condition 2 also reduces to Condition 1 and dependence of the implied constants on m becomes out of consideration.

Local lemma of "Ganelius-Tenenbaum type"

In this section, we prove a new variant of the Lemma of Ganelius, (which is itself a further effective version of an inequality of Bohr). Ganelius' Lemma was used in various versions by Tenenbaum to obtain effective estimations related to the Wiener-Ikehara theorem and other Tauberian theorems. However, as it is seen from the version we quote below, the estimates worked in a "uniform" way, that is, only globally valid estimates could be used in the final results. Our goal will be to relax on this, that is to exploit asymptotic decrease of the estimates in the conditions. Succeeding in that provides a key to our approach in finding effective error terms in the Wiener-Ikehara Theorem even if only asymptotic negligibility conditions are available on the functions to be analyzed. That relaxation opens up the possibility to consider even not necessarily increasing functions A(x), but also those who have only some relaxed asymptotic conditioning on their monotonicity (as discussed in Section 2). Lemma 5.1 (Ganelius-Tenenbaum). Let g : R → R be a bounded function of L 1 (R) satisfying for some positive T

sup x≤y≤x+1/T (g(y) -g(x)) ≤ K < ∞.
Then we have

g ∞ ≤ 16K + 6 T -T | g(τ )|dτ.
For a proof, see e.g. [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF], p. 328. Lemma 5.2 (Ganelius type lemma). Let Φ be a nonnegative even function on R, nonincreasing on the half-line [0, ∞). Let a > 0 and T > 0 be real numbers and g : R → R be a bounded measurable function satisfying

sup 0<y<1/T {g(x + y) -g(x)} ≤ a + Φ(x) x ≥ x 1 := 8 T 1 + Φ(0) a ( 44
)
and g(τ

) = 0 (|τ | < T ). ( 45
)
Then we have

|g(x)| ≤ 8 14 3 (a + Φ(0))(a + Φ(x/2)) (46)
for all x > x 0 (T, a) := (16/T ) 1 + Φ(0)/a.

Proof. Our proof is a variation of the argument on page 327 of [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF]. Fix a real number x > 0 such that g(x) = 0 and denote θ := g(x)/|g(x)|.

We define χ T as in [START_REF] Whittaker | A course of modern analysis. Fourth[END_REF]. We have supp χ T = [-T, T ], thus g χ T = 0 and g * χ T = 0. Let q > 0 be fixed. By the application of(44) ( 2q + 1)-times, we get

0 = θ ∞ -∞ g(x -t -θq/T )χ T (t)dt = θ |t|>q/T g(x -t -θq/T )χ T (t)dt + θ q/T -q/T g(x)χ T (t)dt -θ q/T -q/T (g(x) -g(x -t -θq/T ))χ T (t)dt ≥ -g ∞ I q + (1 -I q )|g(x)| -( 2q + 1) (1 -I q ) a + max [x-2q/T,x+2q/T ]

Φ

where I q is defined in ( 14), and we used that Condition (44) applies, that is, x -2q/T must exceed x 1 . Using (44), Ganelius' lemma, (contained in Lemma 5.1 above) implies g ∞ ≤ 16(a + Φ(0)).

With q ≥ (π/4 -1/2) -1 (hence πq/4 -1 ≥ q/2 and q ≥ 3), one gets for

x ≥ 4q/T |g(x)| ≤ ( 2q + 1) (a + Φ(x -2q/T )) + 16(a + Φ(0)) πq/4 -1 ≤ 2 + 1 3 q (a + Φ(x/2)) + 32(a + Φ(0)) q . ( 47
)
Now,

32(a + Φ(0)) 2 + 1 3 (a + Φ(x/2)) ≥ 4 6 7 ≥ (π/4 -1/2) -1 ,
so we can choose q = 32(a+Φ(0)) (2+1/3)(a+Φ(x/2)) in ( 47). With such a q and for x > 16 T 1 + Φ(0)/a, we also have x ≥ 4q/T , and in particular x -2q/T ≥ x/2 ≥ x 1 , thus we obtain

|g(x)| ≤ 8 14 3 (a + Φ(0)) (a + Φ(x/2)).
Lemma 5.3 (Tenenbaum type Lemma). Let Φ be a nonnegative even function on R, nonincreasing on the half-line [0, ∞). Let b > 0 and T > 0 be real numbers and g : R → R be a bounded measurable function satisfying

sup 0<y<1/T {g(x + y) -g(x)} ≤ b + Φ(x) x ≥ x 1 := 8 T . ( 48
)
Then, for all x > x 0 (T, b) = (16/T ) 1 + Φ(0)/b, we have

|g(x)| ≤ 20 Φ(0)Φ(x/2) + Φ(0) b + 1 π T -T | g(τ )|dτ + b + 1 π T -T | g(τ )|dτ . ( 49
)
Proof. We follow Tenenbaum's proof of Lemma 5.1, i.e. Théorème 7.15 on page 328 of [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF], replacing the Lemma of Ganelius (see [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF]Théorème 7.14]) by Lemma 5.2 in the argument. So for ε > 0 we define on R the function

α(t) := 2 πεt 2 sin εt 2 sin 2T + ε 2 t
having the Fourier transform

α(τ ) =      1 if |τ | ≤ T T +ε-|τ | ε if T < |τ | ≤ T + ε 0 if |τ | > T + ε
and the functions f := g * α and h := g -f . For any real number x, we obviously have |g(x)| ≤ |h(x)|+ f ∞ , furthermore, thanks to the Condition (48), we also have

sup 0<y<1/T {h(x + y) -h(x)} ≤ 2 f ∞ + sup 0<y<1/T {g(x + y) -g(x)} (50) ≤ 1 π T +ε -T -ε | g(τ )|dτ + b + Φ(x),
where we used

f ∞ ≤ f 1 /(2π) and | f | = | α g| for the last inequality. On the other hand, h = g -f = g(1 -α) = 0 on the interval [-T, T ],
whence Lemma 5.2 can be applied with a

:= b + Q, Q := 1 π T +ε -T -ε | g|. Noting that b ≤ a implies x 0 (T, b) ≥ x 0 (T, a) while x 1 (a, T ) > 8/T , this gives for x ≥ x 0 (T, b), |h(x)| ≤ 8 14 3 (b + Q + Φ(0))(b + Q + Φ(x/2)). Adding |f (x)| ≤ f ∞ ≤ 1 2 Q yields the inequality |g(x)| ≤ 8 14 3 (b + Q + Φ(0))(b + Q + Φ(x/2)) + 1 2 Q ≤ 20 (b + Q + Φ(0))(b + Q + Φ(x/2)).
After letting ε tend to 0, the estimate (49) results by comparing squares of the expressions on the right there and here.

Effective Wiener-Ikehara Theorem

To get a theorem with an effective error term, we shall need a strong restriction on both the decrease of the function A and the regularity of the Mellin-Stieltjes transform on the border line. We first recall the effective Wiener-Ikehara theorem in Tenenbaum's version [8, p.326]. Theorem 6.1 (Effective Wiener-Ikehara). Let A be a nondecreasing function such that the integral A(s) = ∞ 0 e -st dA(t) is convergent for σ > 1. Furthermore, let α ≥ 0 and ω > -1 be constants and let us define the functions

G(s) := 1 s + 1 A(s + 1) - α s ω+1 (51)
and correspondingly

η G (σ, T ) := T -T |G(2σ + iτ ) -G(σ + iτ )| dτ. ( 52
)
If we assume that the above functions (51) and (52) satisfy for any fixed T > 0 that σ ω η G (σ, T ) = o(1) as σ → 0 + , then we have (with an explicit O-constant depending only on α and ω)

A(t) = e t t ω α Γ(ω + 1) + O(r(t)) (t ≥ 1),
where 1) in Theorem 6.1 ensures that r(t) be o(1) when t → ∞. This hypothesis can be weakened or withdrawn but the theorem may thus lead to a weaker or trivial upper bound.

r(t) = inf T ≥64 1 T + 1 t ω η G 1 t , T + 1 (T t) ω+1 . Remark 6.1. The hypothesis σ ω η G (σ, T ) = o(
Note that in case A has a meromorphic continuation on the boundary line, the announced upper bound is nontrivial, since we have the following fact, certainly well-known and folklore, but difficult to trace back as to its origin (c.f. [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF]Exc. 224] for one direction of the statement).

η G (σ, T ) ≤ m-1 k=0 a k σ -k + η H (σ, T ),
where lim σ→0+ η H (σ, T ) = 0 and the coefficients a k can be explicitly computed in terms of the coefficients of the principal part P .

Proof. First let us assume that the meromorphic function G has some pole q = ib, with -T < b < T , of order m ≥ 1. We are to prove that σ m-1 η G (σ, T ) is bounded from below by a positive constant.

There exists α ∈ C such that for K > 0 and ε > 0, we can find δ = δ(K, ε) > 0, p ∈ C * and a function h such that |b ± Kδ| < T , h is analytic on R := [0, 2δ] × [(b -Kδ)i, (b + Kδ)i] and for any s ∈ R, we have G(s) = α(s -q) -m (1 + h(s)) and |h(s)| ≤ ε. For 0 < σ < δ, we therefore have

η G (σ, T ) > b+Kσ b-Kσ |G(2σ + it) -G(σ + it)| dt ≥ |α| Kσ -Kσ (1 -ε)|(σ + it) -m | -(1 + ε)|(2σ + it) -m |dt ≥ 2|α| ((1 -2ε)W m (σ, Kσ) -W m (2σ, Kσ)) ≥ 2|α| σ m-1 (1 -2ε) K 0 du (1 + u 2 ) m/2 -2 -m+1 K/2 0 du (1 + u 2 ) m/2 .
For m ≥ 2 and ε ≤ 1/4, we thus have

η G (σ, T )σ m-1 > |α| K K/2 (1 + u 2 ) -m/2 du,
where the expression on the right hand side is a positive constant, depending only on α, m and K. For m = 1, we have

η G (σ, T ) ≥ 2|α| K K/2 (1 + u 2 ) -1/2 du -2ε K 0 (1 + u 2 ) -1/2 du ≥ 2|α| K 2 √ 1 + K 2 -2ε(1 + ln K) .
This lower bound is a constant, depending only on α, K and ε and strictly positive whenever K ≥ 3 and ε < 1/(8 √ 2 ln K).

If we take now a function H, analytic on [-iT, iT ], then we can choose δ > 0 such that H is analytic on

D := [0, 2δ] × [-iT, iT ] and K > 0 such that |H (s)| ≤ K for s ∈ D. For any σ < δ we get η H (σ, T ) = T -T 2σ σ H (ξ + it)dξ dt ≤ T -T 2σ σ |H (ξ + it)| dξdt ≤ 2T σK,
which obviously tends to zero together with σ. It remains now to calculate the upper estimation of η Q (σ, T ) for a general singular term Q(s) = a(s -ib) -j , where a ∈ C, |b| < T and j ∈ N * , in the principal part P (s) of G(s). We can write

η Q (σ, T ) ≤ ∞ -∞ a (2σ + i(t -b)) j - a (σ + i(t -b)) j dt ≤ |a|jσ ∞ -∞ dt |σ + it| j+1 = 2|a|jσW j+1 (σ, ∞) ≤ 2|a|(j + 1) σ j-1 ,
according to (20) and ( 22). Summing the contributions of the poles, we get an explicit upper bound of the desired form for η P (σ, T ) only depending on the coefficients of the principal part P of G.

Theorem 6.1 apply immediately to Dirichlet series with a pole at s = 1.

Theorem 6.2 (Meromorphic Wiener-Ikehara). Let (a n ) n be a sequence of positive real numbers such that the Dirichlet series A(s) = n≥1 a n n -s is convergent for σ > 1 and has a meromorphic continuation on σ ≥ 1 with one single pole at s = 1, with order m ∈ N and α := lim s→0+ s m A(s + 1).

With the functions G(s) and η G (σ, T ) defined in (51) and (52) above, we have

A(x) := n≤x a n = x(log x) m-1 α (m -1)! + O(ρ(x)) with ρ(x) := inf T ≥64 1 T + 1 log m-1 x η G 1 log x , T .
We get a result similar to Theorem 6.2 also for complex sequences majorized by "good" sequences. Corollary 6.1. Let (a n ) be a sequence of complex numbers and (b n ) be a sequence of positive numbers such that the Dirichlet series A(s) = n≥1 a n n -s and B(s) = n≥1 b n n -s are absolutely convergent for σ > 1 and have meromorphic continuations on σ ≥ 1 with at most one single pole, simple, at s = 1, with residue α and β respectively. We assume further that |a n | ≤ b n and we define the functions

G a (s) = 1 s + 1 A(s + 1) - α s , G b (s) = 1 s + 1 B(s + 1) - β s
and the associated η functions η a and η b the way we have done in (51)-( 52).

Then we have

A(x) := n≤x a n = αx + O(xρ(x)) with ρ(x) = inf T ≥64 1 T + η b 1 log x , T + η a 1 log x , T .
To prove this corollary, it is enough to consider a

n := a n + b n and a

(2) n := a n +b n . This had already been noticed for the non effective Wiener-Ikehara Theorem in [4, p.7] by M.R and V.K Murty and can be found also in the context of the Karamata Theorem in [8, Th. 7.7, page 318]. However, there is no satisfactory description of the class of Dirichlet series with their coefficients admitting a positive majorization by some sequence subject to the Wiener-Ikehara Theorem. Moreover, even if such a majorization is found, from the analytic behavior of the majorant, no consequence as to the analytic behavior of the original Dirichlet series can be drawn. For example, the sequence

a n := 2 cos(log n)Λ(n) (which is the coefficient sequence of the Dirichlet series ζ /ζ(s + i) + ζ /ζ(s -i))
, is clearly majorized by 2Λ(n), the coefficient sequence of 2ζ /ζ, which has no singularities on the 1-line (apart from the simple pole at s = 1), but the original Dirichlet series admits two other singularities at 1±i, too. Therefore, it seems that descriptive formulation of such majorization results is not convenient, and that the case dependent study of the situation (singularities, which majorization to use, etc.) is more appropriate.

We can now state our first main result which gives an effective error term for the asymptotic estimate of a moderately decreasing function. Theorem 6.3. Let A be a real function of locally bounded variation, vanishing on (-∞, 1), such that the Mellin transform A(s) := ∞ 1 x -s dA(x) is convergent for σ > 1. We define the function

G(s) = 1 s + 1 A(s + 1) - n k=0 m k l=1 c k,l (s + ib k ) ω k,l +1 + c k,l (s -ib k ) ω k,l +1 , ( 53
)
where n and m k are positive integers, c k,l are complex numbers, ω k,l and b k are real numbers satisfying b 0 = 0 and b k > 0 if k ≥ 1 and ω k,l > -1. Furthermore, for T > 0 and σ > 0, we also define the function

η(σ, T ) = T -T |G(2σ + iτ ) -G(σ + iτ )| dτ. ( 54
)
Assume that the function A is moderately decreasing on [0, +∞) in the sense of Definition 2.3.

Then for x ≥ x 0 (T ) := max(e 16/15 ; e 32/T ), we have

A(x) -x n k=0 m k l=1 (log x) ω k,l Γ(ω k,l + 1) 2 c k,l e -ib k log x xρ T (x) (55)
where the implicit constant depends only on

B 1 , B 2 , c k,l , Ω := max k,l ω k,l , b k and A -[1,e]
, and where with ω := min k,l ω k,l and Ω := max k,l ω k,l ,

ρ T (x) = ϕ( √ x) + max κ=1/2,1 e 10/T T + 1 η 1 log x , T + (56)
e 10/T T log

Ω + x + 1 log x 1 T Ω+1 + 1 T ω+1 κ ,
with ϕ being the function introduced in (9) in Definition 2.3. Remark 6.2. The upper bound we get only has a chance to be small when T is large. In case T ≥ T 0 > 0 we get an upper bound of the form

ρ T (x) = ϕ( √ x) + max κ=1/2,1 η 1 log x , T + 1 T ω+1 log x + log Ω + x T κ ,
where the implicit constant depends on the previous parameters and on T 0 .

Remark 6.3. Korevaar gives some tauberian theorem for a similar type of functions in [START_REF] Korevaar | Tauberian Theory -a Century of Developments[END_REF], p.381-382 but his result seems less general than ours.

Proof. Our proof combines the arguments of the proof of Proposition III.10.2 in [START_REF] Korevaar | Tauberian Theory -a Century of Developments[END_REF] and of the proof of Théorème II.7.13 in [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF] with our Lemma 5.3. We define for t ∈ R and σ ∈ (0, 1), the functions

g σ (t) := h σ (t) -h 2σ (t) = h σ (t)(1 -e -σt ) = A(e t )e -(1+σ)t (1 -e -σt )
and with β(ω, t) defined in (25) also

L σ (t) := g σ (t) -2 n k=0 m k l=1 c k,l e -ib k t σ -ω k,l β(ω k,l , σt).
We shall use Lemma 5. The Fourier transform of g σ is

g σ (τ ) = A(σ + iτ + 1) σ + iτ + 1 - A(2σ + iτ + 1) 2σ + iτ + 1 = G(σ + iτ ) -G(2σ + iτ ) + n k=0 m k l=1 c k,l 1 (σ + i(τ + b k )) ω k,l +1 - 1 (2σ + i(τ + b k )) ω k,l +1 + c k,l 1 (σ + i(τ -b k ) ω k,l +1 - 1 (2σ + i(τ -b k ) ω k,l +1 , hence those of L σ is L σ (τ ) = G(σ + iτ ) -G(2σ + iτ ). Therefore T -T | L σ (τ )|dτ = η(σ, T ). (57)
For any b ∈ R and -1 < ω ≤ 0 we can infer

1 (σ + i(τ ± b)) ω+1 - 1 (2σ + i(τ ± b)) ω+1 ≤ (ω + 1)σ |σ + i(τ ± b)| ω+2 ,
expressing the difference by an integral, and so (19), (20) and ( 22) leads to

T -T 1 (σ + i(τ ± b)) ω+1 - 1 (2σ + i(τ ± b)) ω+1 dτ ≤ 2(ω + 2)σ -ω ≤ 6σ -ω ,
while for any b ∈ R and ω > 0 we integrate the terms separately and apply (19), (20) and ( 22) with m = ω + 1 > 1 giving the estimate ≤ 2(1 + 2 -ω ) ω+1 ω σ -ω ≤ 6σ -ω for the integral on the left hand side. So in all cases we have obtained an estimate by 6σ -ω . Integrating the above expression for g σ (τ ) and using this estimate with ω = ω k,l and b = b k leads to

T -T | g σ (τ )|dτ ≤η(σ, T ) + n k=0 m k l=1 |c k,l | 12 σ -ω k,l . (58)
Next we give a lower bound for the difference L σ (u+v)-L σ (u), hence of g σ (u + v) -g σ (u), for arbitrary values of σ ∈ (0, 1), u ∈ R and 0 ≤ v ≤ 1/T . Since A vanishes on (-∞, 1), we assume u + v ≥ 0. For σ ∈ (0, 1), u ∈ R and 0 ≤ v ≤ 1/T , we have

g σ (u + v) -g σ (u) = F 1 (u, v) + g σ (u)F 2 (u, v) with F 1 (u, v) = e -(1+σ)(u+v) 1 -e -σ(u+v) A e u+v -A (e u ) F 2 (u, v) = e -(1+σ)v e -σu -e -σ(u+v)
1 -e -σu + e -(1+σ)v -1.

• First, we give a lower bound for g σ (u)F 2 (u, v).

-For u < 0, g σ (u)F 2 (u, v) obviously vanishes together with A(e u ) and g σ (u), thus g σ (u)F 2 (u, v) = 0.

-Note that in case u = 0, F 2 (u, v) is not defined but

g σ (u)F 2 (u, v) = A(1)e -(1+σ)v 1 -e -σv ≥ -σv A -[1,e] .
-For any σ ∈ (0, 1), u > 0 and v ≥ 0, we have

F 2 (u, v) = e -(1+σ)v 1 -e -σv e σu -1 + e -(1+σ)v -1, thus -2v ≤ -(1 + σ)v ≤ -(1 -e -(1+σ)v ) ≤ F 2 (u, v) ≤ 1 -e -σv e σu -1 ≤ v u ,
and therefore

F 2 (u, v)g σ (u) > -2v g σ ∞ if A(e u ) ≥ 0. If A(e u ) < 0 and u ≥ 1, we use F 2 (u, v) ≤ v, thus F 2 (u, v)g σ (u) > -v g σ ∞ .
In case 0 < u < 1 and A(e u ) < 0, we write

F 2 (u, v)g σ (u) = A(e u )(f (u+v)-f (u)) ≥ -{A(e u )} -|f (u+v)-f (u)| with f (u) = e -(1+σ)u (1 -e -σu ). Since sup u≥0 |f (u)| ≤ 1 + σ ≤ 2, we get F 2 (u, v)g σ (u) ≥ -2v{A(e u )} -≥ -2v A -[1,e] .
Therefore, for u ∈ R, σ ∈ (0, 1) and v ≥ 0, we always have

F 2 (u, v)g σ (u) ≥ -2v A -[1,e] + g σ ∞ .
• To deal with the first term F 1 (u, v), we use [START_REF] Widder | The Laplace transform[END_REF].

-We therefore get for σ ∈ (0, 1), u ≥ 0 and v ≥ 0

F 1 (u, v) ≥ -{F 1 (u, v)} -= -(1 -e -σ(u+v) )
{A(e u+v ) -A(e u )} - e (1+σ)(u+v) ≥ -e -(u+v) {A(e u+v ) -A(e u )} -

≥ -B 1 (1 -e -v ) -B 2 ϕ(e u )e -v ≥ -B 1 v -Φ(u), with Φ(u) = B 2 ϕ(e u ).
-For σ ∈ (0, 1), u < 0 and u + v ≥ 0, we use A(e u+v ) -A(e u ) =

A(e u+v ) = A(e u+v ) -A(0) and get

F 1 (u, v) ≥ -(1 -e -σ(u+v) ) {A(e u+v )} - e (1+σ)(u+v) ≥ -(1 -e -σ(u+v) ) e -(u+v) {A(e u+v ) -A(0)} - ≥ -(1 -e -σ(u+v) ) B 1 + B 2 e -(u+v) ≥ -(B 1 + B 2 )(1 -e -σ(u+v) ) ≥ -(B 1 + B 2 )σ(u + v) ≥ -(B 1 + B 2 )v since 0 ≤ u + v ≤ v and σ < 1.
For σ ∈ (0, 1), u ∈ R and v ≥ 0, the above lower estimates yield

g σ (u + v) -g σ (u) ≥ -2v g σ ∞ -B v -Φ(u), (59)
where

B = B 1 + B 2 + 2 A -[1,e]
and Φ is defined by Φ(u) := B 2 ϕ(e |u| ).

We shall now use Lemma 3.1 to give an upper bound for g σ ∞ .

Since A is moderately decreasing, [START_REF] Wiener | Tauberian theorems[END_REF] provides for any real number u the estimate A(e u )e -u ≥ -λ 3 with λ

3 = B 1 + B 2 . Consequently, g σ (u) ≥ -λ 3 .
Let λ 2 be a positive fixed constant and define as in the proof of Theorem 4.1, S 0 (λ 2 ) := {u ∈ R : A(e u ) > λ 2 e u }.

Note that here, we do not assume u ≥ 0. Nevertheless, for u < 0, we have A(e u ) = 0 thus obviously u / ∈ S 0 (λ 2 ) and also g σ (u) = 0. Therefore, if u ∈ S 0 (λ 2 ), then g σ (u) ≤ λ 2 both for u < 0 and for u ≥ 0, too.

We now choose λ 2 = 5 max(B 1 , B 2 )e 10/T as before (preceding (36)). Then using the first inequality of (36), we get for u ∈ S 0 (λ 2 ) \ {0}, v ∈ [0, 10/T ] and 0 < σ < σ 0 := min 1, log 2 10 T ,

g σ (u + v) = (1 -e -σ(u+v) )h σ (u + v) ≥ (1 -e -σ(u+v) ) 4 5 e -(1+σ)10/T h σ (u) = 4 5
e -σ10/T e -10/T 1 -e -σ(u+v)

1 -e -σu g σ (u) ≥ 2 5 e -10/T g σ (u).

Note that the estimate remains valid when u = 0. Now we can apply (the first part of) Lemma 3.1 similarly as in the proof of Thus with (59), we get for 0 < σ < σ 0 , u ∈ R and 0

< v < 1/T g σ (u + v) -g σ (u) ≥ - e 10/T T 13.2 n k=0 m k l=1 |c k,l |σ -ω k,l (60)
+ 10λ 3 + 1.2η(σ, T ) - B T -Φ(u).
We shall now give a lower bound for the difference L σ (u + v) -L σ (u). For any b, γ ∈ R and -1 < ω Lemma 3.2 yields with y :

= σv ∈ [0, 1/T ] |β (ω, σ(u + v)) cos(b(u + v) + γ) -β(ω, σu) cos(bu + γ)| ≤ |β(ω, σ(u + v)) -β(ω, σu)| + | cos(b(u + v) + γ) -cos(bu + γ)| • |β(ω, σu)| ≤ 2 √ π σ T + σ T ω+1 + |b| T .
Applying this with γ := arg c k,l , b := b k and ω := ω k,l gives

L σ (u + v) -L σ (u) ≥ g σ (u + v) -g σ (u) (61) - 4 √ π n k=0 m k l=1 |c k,l | σ T + σ T ω k,l +1 + |b k | T σ -ω k,l .
Taking (60) into account, a calculation gives for all u ∈ R, 0 ≤ v ≤ 1/T and 0 < σ ≤ σ 0

L σ (u + v) -L σ (u) ≥ - C 3 (σ, T ) T -Φ(u) - 4 √ π σ n k=0 m k l=1 |c k,l | 1 T ω k,l +1 with C 3 (σ, T ) := e 10/T (1.2η(σ, T ) + 10λ 3 ) + B + n k=0 m k l=1 |c k,l |(13.2e 10/T + 4 √ π (σ + |b k |))σ -ω k,l .
Finally, we apply Lemma 5.3 to -L σ . Since C 3 (σ, T ) > 10λ 3 ≥ 10B 2 = 10Φ(0), the validity of the estimate (49) can be assured for t ≥ 16 T 1 + T /10, hence for t ≥ t 0 := 32 max(1/30; 1/T ) > 1/σ 0 . Therefore, Lemma 5.3 and (57) gives for such t that

|L σ (t)| ≤20 C 3 (σ, T ) T + 4 √ π σ n k=1 m k l=1 |c k,l | 1 T ω k,l +1 + 1 π η(σ, T ) + 20 B 2 C 3 (σ, T ) T + 4 √ π σ n k=1 m k l=1 |c k,l | 1 T ω k,l +1 + 1 π η(σ, T ) + 20 B 2 Φ(t/2).
We choose σ = 1/t and x = e t and assume x ≥ x 0 (T ) := e t 0 = max(e 16/15 ; e 32/T ) (hence σ < σ 0 ) to obtain

A(x) -x n k=0 m k l=1 2 c k,l e -ib k log x (log x) ω k,l Γ(ω k,l + 1) ≤ xρ T (x) with ρ T (x) = 20e 1 -e -1 B 2 ϕ( √ x) + R T (x) + B 2 R T (x) (62)
where

R T (x) = C 4 (T ) T + C 5 (T )η 1 log x , T + 1 T n k=0 m k l=1 C (1) k,l (log x) ω k,l + C (2) k,l log x (log x) ω k,l + 1 T ω k,l , C 4 (T ) = 10λ 3 e 10/T + B , C 5 (T ) = 1.2e 10/T T + 1 π , C (1)
k,l (T ) = |c k,l | 13.2e 10/T + 4 √ π |b k | , C (2)
k,l = 4 √ π |c k,l |.
Remark 6.4. Observe that in case B 2 = 0 can be taken, the relative losses of taking squareroots in (62) vanish, and we get the sharper estimate with

ρ T (x) = e 10/T T + 1 η 1 log x , T + e 10/T T log Ω + x + 1 log x 1 T Ω+1 + 1 T ω+1
in place of (56) in the statement of the theorem. Corollary 6.2. Under the same conditions then in Theorem 6.3, but assuming (instead of the condition of A(x) moderately decreasing) that A(x) is nonnegative and nondecreasing, we obtain the bound (55) with the error function

ρ T (x) = e 10/T T + 1 η 1 log x , T + e 10/T T log Ω x + 1 log x 1 T Ω+1 + 1 T ω+1 .
(63) In particular, when T ≥ 1, we find (55) to hold with

ρ T (x) = η 1 log x , T + log Ω x T + 1 log x T 1+ω . (64)
Proof. Compared to Remark 6.4 the only change we need to explain is the use of log Ω x in place of log Ω + x. But when A is nondecreasing, then not only B 2 = 0, but also we have the condition (9) even with B 1 = 0. Moreover, from nonnegativity of A it also follows that A 1 := A -[1,e] = 0, so also B = 0 and in the last formula of the proof of Theorem 6.3 giving R T (x), we find C 4 (T ) = 0, too. Therefore, no term of the order 1/T appears, and instead of 1/T + log Ω x/T log Ω + x/T we can as well write log Ω x/T . When we even have T ≥ 1, (63) clearly entails (64), too.

Remark 6.5. Observe that the second part of Corollary 6.2 clearly covers the original result Theorem 6.1 of Tenenbaum.

A similar result can also be derived for functions satisfying the slowly decreasing condition, which is a more standard condition on the controlled decrease of functions. Proof. Our proof is very similar to the proof of Theorem 6.3, so we use the same notations. As the control on the decrease was not used in that, we obtain the same upper bounds for the integrals 1 -e -σu -1 = e -σv -1 + e -σv 1 -e -σv e σu -1 .

We choose now λ 2 := 50ν(e 1/T ) + A [1,e] and consider S 0 (λ 2 ) := {u ∈ R : A(e u ) > λ 2 e u }.

Let u ∈ R be such that g σ (u) > λ 2 . Then u > e and u ∈ S 0 (λ 2 ) and for 0 ≤ v ≤ 10/T we have A(e u+v ) e u+v -

A(e u ) e u ≥ -ν(e 10/T ) ≥ -10ν(e 1/T ) ≥ -

1 5 λ 2 .
Therefore, for any u ∈ R with g σ (u) > λ 2 , and for any 0 ≤ v ≤ 10/T, 0 < σ ≤ σ 0 = min(1, log 2 10 T ), we have g σ (u + v) g σ (u) = A(e u+v )/e u+v A(e u )/e u • e -σ(u+v) (1 -e -σ(u+v) ) e -σu (1 -e -σu ) ≥ 1 -λ 2 /5 A(e u )/e u • e -σv ≥ 4 5

e -10σ 0 /T ≥ 2 5 .

To estimate g σ ∞ , we next use the first part of Lemma 3.1 with λ 1 = 2/5, and the above chosen λ 2 and λ 3 , together with (58). We get That concludes the proof.

The above result has a point only for T large, in particular when T can be fixed arbitrarily large. For T ≥ 1 the theorem gets a slightly simpler form as follows. order of A(x) is to be x log Ω + x, anyway. But in this case starting out even from an increasing function A(x) it is not clear, how slow decrease of A(x)/x may follow, if at all, while A(x) is still of moderate decrease and Theorem 6.3 still works. To apply Theorem 6.4, one may have to consider the order of η, or other information about the behavior of A(x). Alternatively, at all probability an analogous theorem, under the assumption that A(x)/x log m x with m = Ω + is slowly decreasing, can similarly be proved, similarly as we have done in case of Theorem 4.1. That would directly work here after an application of Proposition 2.1 with (x) := x log Ω + x.

We spare the reader from these calculations, for in all cases a direct use of Theorem 6.3 with the increasing, or moderately decreasing properties of A(x) would give a simpler solution. However, let us note that such a result would be slightly more general (as slow decrease of A(x)/x would imply that of A(x)/x log Ω + x, too, and as also the moderate decrease of A(x) implies this condition -assuming in both cases A(x) = O(x log Ω + x)). Compare Remark 4.1.

F

  which is already independent of y ∈ [x, λx]. Taking lim (x)/(x (x)) is slowly decreasing.

Corollary 4 . 1 .

 41 Under the same conditions than in Theorem 4.1 but replacing (29) by the fact that the Mellin-Stieltjes transform A has a meromorphic continuation in a neighborhood of s = 1, with a pole of order m ∈ N (i.e., a regular point if m = 0) at s = 1, we have A(x) = O(x log (m-1) + x) for all x ≥ e.

Lemma 4 . 1 .

 41 Let A : R → R be a function of locally bounded variation vanishing on (-∞, 1) and assume that the Mellin-Stieltjes transform A(s) := ∞ 1

Corollary 4 . 2 .

 42 Assume the same hypothesis as in Theorem 4.1, but replacing (29) by η

Proposition 6 . 1 .

 61 Let T be a non negative real number and G be a meromorphic function in some open domain containing the closed interval [-iT, iT ] with -iT and iT regular points of G. With η G (σ, T ) as above in (52), the meromorphic function G has poles of order at most m in (-iT, iT ), if and only if σ m-1 η G (σ, T ) = O(1) as σ → 0+. In particular G has no singularities on the interval [-iT, iT ] if and only if lim σ→0+ η G (σ, T ) = 0. Furthermore, if H = G -P where P is the principal part of the function G on [-iT, iT ], then we can write

  3 to give an upper bound for the function L σ . Let T > 0. We shall need an upper bound for the integral T -T | L σ (τ )|dτ hence for the integral T -T | g σ (τ )|dτ and a lower bound for the difference L σ (u + v) -L σ (u), hence of g σ (u + v)g σ (u), for arbitrary values of σ ∈ (0, 1), u ∈ R and 0 ≤ v ≤ 1/T .

Theorem 4 .

 4 1 and with λ 1 := 2 5 e -10/T and λ 2 , λ 3 chosen above. Taking into account (58) we infer g σ ∞ ≤ e 10/T 5λ 3 + 0.55 T -T | g σ (τ )| dτ ≤ e 10/T 5λ 3 + 0.6η(σ, T ) + 6l |σ -ω k,l .

Theorem 6 . 4 .log x + 1 T

 641 Let A(x) be a real function of locally bounded variation, vanishing on (-∞, 1), such that the Mellin transform A(s) := ∞ 1 x -s dA(x) converges for σ > 1. As in Theorem 6.3, we define the functions G(s) and η G (σ, T ) according to (53) and (54), respectively, where n and m k are positive integers, c k,l are complex numbers, ω k,l and b k are real numbers satisfying b 0 = 0 and b k > 0 if k ≥ 1, and ω k,l > -1, T > 0 and σ > 0.Further we assume that A(x)/x is slowly decreasing on [1, ∞) in the sense of Definition 2.2, and that with Ω := max k,l ω k,l we haveη(σ, T ) = O(σ -Ω + ) (0 < σ < 1). (65)Then for x ≥ max(e; e 17/T , e 17/ √ T ) the error formula (55) holds withρ T (x) = Ψ e 1/T (1)Ψ e 1/T ( √ x) + R T (x) + Ψ e 1/T (1)R T (x),(66)where with ω := min k,l ω k,lR T (x) =ν AΩ+1 log x (67)with ν, ν and Ψ being the functions introduced in (2), (3) and (4), respectively, and where the implicit constants only depend on ν(A(x)/x; e), A[1,e] , A(2), the constant in (65), c k,l and b k .

|

  g σ (τ )|dτ, while we need to get a new lower bound for the difference g σ (u + v) -g σ (u).We now write for σ ∈ (0, 1), u ∈ R and0 ≤ v ≤ 1/T g σ (u + v) -g σ (u) = F 3 (u, v) + g σ (u)F 4 (u, v)withF 3 (u, v) := e -σ(u+v) 1 -e -σ(u+v)A (e u+v ) e u+v -A (e u ) e u , F 4 (u, v) := e -σv 1 -e -σ(u+v)

| 1 T

 1 g σ (τ )|dτ + max(λ 2 , λ 3 ) ≤ 50ν(e 1/T ) + A [1,e] + λ 3 + 0.6η(σ, T ) l |σ -ω k,l .(78)In turn, this estimate can now be substituted into (72) resulting ing σ (u + v) -g σ (u) ≥ -ν(e 1/T ) -Φ(u) -1 T 50ν(e 1/T ) + A [1,e] + 2M σ + λ 3 + 0.6η(σ, T ) + 6l |σ -ω k,l .Substituting this into (61) leads, for σ ∈ (0, σ 0 ), u ∈ R and 0 < v < 1/T toL σ (u + v) -L σ (u) ≥ -Φ(u) -ν(e 1/T ) -C 6 (σ, T ) l | T ω k,l +1 , with C 6 (σ, T ) := 50ν(e 1/T ) + A [1,e] + λ 3 + 2M σ + 0.6η(σ, T ) + n k=0 m k l=1 |c k,l | 4 √ π (σ + |b k |) + 6.6 σ -ω k,l .Finally, we apply Lemma 5.3 to -L σ , with Φ the even function occurring here and the role of b played by the rest of the lower estimate for the difference of L σ (u + v) -L σ (u). Note that then b > 50ν(e 1/T )/T ≥ 50{ν(e 1/T ) + Φ(0)}/T ≥ 50Φ(0)/T , whence x 0 (T, b) ≤ (16/T ) max( √ 1.02, √ 1.02T ) < max(17/T, 17/ √ T ). It follows from Lemma 5.3, (49) and (57) that for t > max(17/T, 17/ √ T )|L σ (t)| ≤ 20 Φ(0)Φ(t/2) + Φ(0) b + 1 π η(σ, T ) + b + 1 π η(σ, T ) ,whenever 0 < σ < min(1, log 2 10 T ). Choosing in this estimate t := 1/σ and x := e t , we get for x ≥ max(e, e 17/T , e 17/ l e -ib k log x (log x)ω k,l Γ(ω k,l + 1) ≤ xρ T (x) with ρ T (x) = 20e 1 -e -1 Ψ e 1/T (1)Ψ e 1/T ( √ x) + R T (x) + Ψ e 1/T (1)R T (x)where R T (x) = ν(e 1/T (e) + 8|A(1)| + K) log min(1,Ω + ) x + log(2 log x) T ω k,l + log ω k,l x k | + 6.6 log ω k,l x .

Corollary 6 . 3 .

 63 Let A(x) be a function satisfying the same hypothesis as in Theorem 6.4 and let T ≥ 1. Define as previously the functions G and η(σ, T ) (0 < σ < 1). Then for x ≥ max(e, e 17/ √ T ), we haveA(x) -x n k=0 m k l=1 (log x) ω k,l Γ(ω k,l + 1)2 c k,l e -ib k log x xρ T (x)

  3, Theorem III.10.1, p. 142 and Proposition III.10.2, p. 143] some convergence and boundedness theorems involving general Tauberian conditions. These results are of two kinds. If the Mellin-Stieltjes transform of A is regular enough throughout the boundary, we get a convergence theorem, whereas if we have local regularity of the Mellin-Stieltjes transform near s = 1, we only get a boundedness theorem. In this section we work out a more general "local theorem" allowing the Mellin-Stieltjes transform to have some singularities at s = 1.

	according to Definition 2.1,
	3. or A satisfies the moderate decrease condition of Definition 2.3 on
	[1, +∞).

Theorem 4.1. Let m ≥ 0 and T > 0 be fixed parameters and let A : R → R be a function of locally bounded variation vanishing on (-∞, 1) such that 1. either we have that A(x)/x is of bounded decrease on [1, ∞) according to Definition 2.1, 2. or we have that A(x)/(x log (m-2) + x) is of bounded decrease on [e, ∞)

* Supported in part by the Hungarian National Foundation for Scientific Research, Project #s K-81658 and K-100461. Work done in the framework of the project ERC-AdG 228005.

MSC 2000

For any σ ∈ (0, 1), u > 0 and v ≥ 0 we immediately have

For σ ∈ (0, 1), u > 0 and 0 ≤ v ≤ 1/T we get from (4)

For simplicity we will write ν(•) for ν(A(x)/x, •) and ν(•) for ν(A(x)/x, •) and we define the function Φ by Φ(u) := Ψ e 1/T (e |u| ). Then for any σ ∈ (0, 1), u ≥ 1 and 0 ≤ v ≤ 1/T , (68) and (69) entail the lower estimate

In case σ ∈ (0, 1), 0 < u ≤ 1 and 0 ≤ v ≤ 1/T , we estimate g σ (u)F 4 (u, v) from below together. The only interesting cases are when g σ (u)F 4 (u, v) is negative, hence when either g σ (u) > 0 and 0 > F 4 (u, v) > -v, or when g σ (u) < 0 and 0 < F 4 (u, v) ≤ v/u. In the first case the combined estimate is still as good as g σ (u)F 4 (u, v) ≥ -v g σ ∞ ≥ -1 T g σ ∞ , so (70) remains valid. In the second case we make use that A(x)/x is slowly decreasing, hence in particular also boundedly decreasing, and thus according to (6) it satisfies A(e u )/e u > -M (u + 1), where M := M (A, 1). So in case 0 < u ≤ 1 we are led to

and so when g σ (u) < 0 and

From this and (70) we thus conclude that for any 0 < σ ≤ 1, u > 0, 0 < v ≤ 1/T , the estimate

holds true. Furthermore, we found for 0 < σ ≤ 1 and u > 0 the lower estimate

In case -1 < Ω < 1 and u > e, this lower bound has to be sharpened. We invoke Corollary 4.2 with m := Ω + 1 under Condition 1 listed in Theorem 4.1. We get for u ≥ e

remain valid for all Ω, so we are led to the lower estimate

valid for all 0 < σ < 1 and u ≥ 0, (so by g σ (u) = 0 (u < 0), for u ∈ R, too). Writing

and taking (73) and (75) into account, we get for all 0 < σ < 1 and u ∈ R

with the implicit constants only depending on ν(e), c k,l , ω k,l , b k , A(2), A [1,e] and the implicit constant occurring in (65), and with

where with ω := min k,l ω k,l and Ω := max k,l ω k,l

Let us give a small discussion of the last result. On the one hand it uses the very condition of "A(x)/x is slowly decreasing", the condition appearing already in the earliest proofs of the Wiener-Ikehara Theorem. On the other hand it does not cover by itself this result, for we did not assume that strong assumptions on the boundary function. That slight shortcoming is remedied by the fact that for A(x) increasing, we have the needed asymptotic evaluation anyway by Theorem 6.3.

The classical proof of the Wiener-Ikehara Theorem was worked out in the case when we have k = 0, m 0 = 0, that is only one term in the singular asymptotic, with ω 0,0 = 0: A(s + 1) ∼ c/s. Then the argument starts with proving that once A(x) is increasing, we as well have A(x) = O(x), a statement fully contained in Corollary 4.1 if the Laplace transform has a meromorphic continuation at s = 1. This can easily be proved also under other, less stringent conditions like e.g. a continuous, or an L 1 boundary function on [1 -iT, 1 + iT ] with some T > 0.

Here in Theorem 6.4 we settled with the more general condition of η(σ, T ) = O(1) (σ → 0) with some T > 0, and thus with the slightly less precise bound A(x) = O(x log log x), coming from the general (lower) estimation of slowly decreasing functions [START_REF] Schmidt | Über divergente Folgen und lineare Mittelbindungen[END_REF]. On the other hand for the special case when A(x) is increasing, we can as well use that A(x) is moderately decreasing, and apply Theorem 6.3, which already concludes, from only boundedness of η, the estimate A(x) = O(x). (Note that in general A(x)/x being slowly decreasing need not entail that A(x) be moderately decreasing, so even the less increasing, as discussed in Section 2.) But once we know A(x) = O(x), we then can conclude from A(x) being increasing, (and also if only A(x) is slowly decreasing, see Corollary 2.1), that also A(x)/x is slowly decreasing, and thus also Theorem 6.3 can be applied. However, this is kind of superfluous once Theorem 6.3 have already been used anyway: the two give very similar error bounds (with the former being slightly better in view of the occurrence of log log x in the latter).

If for the application of these results we have a situation where Ω > 0, then the slight loss regarding the occurrence of log log x diminishes and the


Aymeric Lamorlette

Quantification of ignition time uncertainty based on the classical ignition theory and Fourier analysis

Keywords: Heat transfer, Piloted ignition, Ignition uncertainty Transfert thermiques, Ignition pilotée, Incertitude de l'ignition

This study aims at modeling the effect of incoming heat flux fluctuations, on solid material ignition. In order to propose a general methodology based on the classical ignition theory that can be applied to any kind of solid target, kernels accounting for the target temperature response regarding an incoming heat flux are considered for thermally thick and thin solids with low or high thermal inertia. A Fourier decomposition of the incoming heat flux is then used to calculate the target response to harmonic heat fluxes. Finally, effects of harmonic fluctuations on ignition are discussed based on the previous analytical results, allowing to discriminate situations where ignition time is expected to be rather predictable from situations where ignition time is expected to be less predictable thanks to an uncertainty quantification of the ignition time.

Introduction

In the frame of fire safety applications, estimation of flaming ignition time of solid material is a major issue. A first approximation, often called the classical ignition theory, consists in considering ignition as two separated mechanisms: Heat and pyrolysis of the solid followed by chemical reactions in the gas phase. If the pyrolysis gas flow is low with normal oxygen concentration and suggesting that ignition occurs when the solid surface temperature reaches the pyrolysis temperature, the ignition time corresponds to the time needed by the solid to heat until it begin its pyrolysis. Indeed, chemical time and mixing time are typically much smaller than pyrolysis time for piloted flaming ignition. These assumptions allow to model ignition as a temperature rise process in the solid material as suggested by [START_REF] Fernandez-Pello | Combustion Fundamentals of Fire[END_REF]. Ignition is thus modeled as a one-dimensional heat conduction problem as proposed by [START_REF] Torero | The SFPE Handbook of Fire Protection Engineering[END_REF] for thermally thick surfaces and as a zerodimension heat transfer problem for thermally thin particles as suggested in [START_REF] Quintiere | Fundamentals of Fire Phenomena[END_REF]. In order to mimic more realistic fire scenarios, an improvement of this theory is proposed in [START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF], where the radiant heat flux is suggested to be linearly time depending, as suggested by [START_REF] Cohen | Relating flame radiation to home ignition using modeling and experimental crown fires[END_REF] to take into account the time-varying heat flux on a stationary target exposed to a spreading fire front. Using this approach, the global trend of the incoming radiant heat flux can be taken into account. However, turbulent motions are common in the fire front as explained by [START_REF] Morvan | Physical phenomena and length scales governing the behaviour of wildfires: A case for physical modelling[END_REF] and often generate periodic or quasi-periodic flame behavior as demonstrated by [START_REF] Atkinson | Fire driven flow in an inclined trench[END_REF] and [START_REF] Dupuy | The effects of slope and fuel bed width on laboratory fire behavior[END_REF], thus periodic or quasi-periodic fluctuations of the radiant heat flux can occur that are not taken into account in the previously described modeling. These fluctuations could be responsible for ignition or fire propagation unpredictabilities since they are not controlled in practical applications: Recent experimental results on fire propagation by [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds; part I: Correlations and observations[END_REF] demonstrate the existence of quasi-periodic fluctuations of the heat flux involving flaming contact on the solid particles composing a fuel layer which seems to be of great matter in particle ignition. Moreover, the study of [START_REF] Morandini | Fire spread experiment across Mediterranean shrub: Influence of wind on flame front properties[END_REF] demonstrates the effect of wind fluctuations on fire propagation, involving processes similar to those suggested in [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds; part I: Correlations and observations[END_REF].

Therefore, this study aims at modeling the effect of heat flux harmonic variations on different solid target of interest, to mimic the effect of the previously described periodic or quasi-periodic processes and investigate ignition uncertainty. The latter is defined as the harmonic flux effect on the temperature rise. It can nevertheless be noticed that if chemical and mixing time were to be considered, uncertainties due to chemistry and mixing should be added to the uncertainty due to the temperature rise, particularly when addressing ignition of the smaller targets, according to [START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF]. This aspect is however not investigated in this study. The suggested target could be PMMA and wood slabs, insulating foams, excelsior or pine needles, which can be classified as thermally thick or thin, with low or high thermal inertia, according to the classification suggested in [START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF]. Hence ignition is modeled for these solid categories. Solutions are discussed for practical fire safety applications, thus use of "practical" in this article will refer to material thermophysical properties of PMMA, wood and insulating foams, with ignition time ranging from a few seconds to a few hundred seconds. The limit (regarding the particle size) between thermally thin and thermally thick material is set according to [START_REF] Benkoussas | Modelling thermal degradation of woody fuel particles[END_REF], where a radiative Biot number, depending on the incoming radiant heat flux, is considered. Indeed, a classical Biot number cannot fully account for the thermal behavior transition since heating is here due to radiative heat transfer.

In order to illustrate the incoming heat flux harmonic variation effect, an example considering the ignition under heat fluxes composed of a constant and a harmonic part is presented. Nevertheless, the methodology allows to extend the study to heat fluxes composed of any kind of slow time-varying variations part adding any harmonic part.

Mathematical formulation: The classical ignition theory

This section recalls the work of [START_REF] Fernandez-Pello | Combustion Fundamentals of Fire[END_REF][START_REF] Torero | The SFPE Handbook of Fire Protection Engineering[END_REF][START_REF] Quintiere | Fundamentals of Fire Phenomena[END_REF][START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF][START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF], using a particular formalism which provides a simple expression for the target temperature response to a Fourier decomposition of the incoming heat flux, and suggesting a slight difference for low thermal inertia solids. Indeed, the study of [START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF] considers series expansion around t → ∞ whereas this study considers a higher series expansion (in comparison with the high thermal inertia case) around t → 0.

For thermally thick targets, a one-dimensional semi-infinite inert solid material is considered with one side exposed to an incoming heat flux Φ(t). The heat conduction equation is solved for the solid target, introducing θ(x, t) = T (x, t) -T 0 where T (x, t) is the solid temperature and T 0 the initial temperature of both the solid and its surrounding air. λ, ρ, Cp and h represent respectively the solid heat conductivity, its density, its specific heat and a total heat transfer coefficient.

For high thermal inertia solids, the associated surface heat loss can be neglected as shown in [START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF][START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF] whereas it is taken into account for low thermal inertia solids. Hence, kernel K is provided by the classical solution suggested in [START_REF] Torero | The SFPE Handbook of Fire Protection Engineering[END_REF][START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF] for high thermal inertia solids. Since the study focuses on the surface temperature evolution, the kernel K is calculated at x = 0, leading to:

K(0, t) = 2 √ t √ πλρCp . ( 1
)
Now, considering an arbitrary function Φ(τ ) as series of steps and using the heat diffusion equation linearity, the response θ(0, t) can be expressed as a sum of response with time offsets for an arbitrary kernel K 0 :

θ(0, t) = t 0 dΦ dτ K 0 (0, t -τ )dτ . ( 2
)
This result is similar to the solution suggested by [START_REF] Carslaw | Conduction of heat in Solids[END_REF] for the heat diffusion equation with K 0 (0, t) = K(0, t).

For low thermal inertia solids, a kernel K H (0, t) that take into account the surface heat losses through a total heat transfer coefficient h is provided, according to [START_REF] Torero | The SFPE Handbook of Fire Protection Engineering[END_REF]:

K H (0, t) = 1 h 1 -exp h 2 λρCp t erfc h √ λρCp √ t . ( 3
)
The practical use of this kernel is however limited due to its mathematical form. Nevertheless it can be simplified assuming that even for material with low thermal inertia, the characteristic time τ KH = λρCp/h 2 of the kernel K H (0, t) is always larger than the ignition time, allowing series expansion of the kernel around t → 0. The latter is then re-written as

K H (0, t) = K(0, t) + K h (0, t) with K h (0, t) ≈ - ht λρCp + 4 3
h 2 t 3/2 √ π (λρCp) 3/2 . ( 4
)
Application of this simplified kernel to a linear time-dependent heat flux does not strictly allow recovering the series expansion result of [START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF] since the series expansion has been made before calculating the convolution. The solution suggested here is nevertheless more accurate regarding the exact solution than the solution proposed in [START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF] which has been experimentally validated, thus validating the simplified kernel. Indeed, in most practical cases, the ignition time is lower than the kernel characteristic time and the series expansion provides an efficient approximation.

In order to assert this assumption, the characteristic time τ KH is calculated for typical material as wood, polymer and insulating foam. Thermophysical properties of these material are extracted from [START_REF] Mowrer | An analysis of effective thermal properties of thermally thick materials[END_REF][START_REF] Demharter | Polyurethane rigid foam, a proven thermal insulating material for applications between +130C and -196C[END_REF][START_REF] Drysdale | An Introduction to Fire Dynamics[END_REF][START_REF] Tihay | Contribution expérimentale et théorique pour la modélisation de la combustion dans les feux de forêts[END_REF]18] and are listed in table 1. In order to estimate the total heat transfer coefficient, different approaches can be suggested. For instance, in [START_REF] Simeoni | Flammability studies for wildland and wildland-urban interface fires applied to pine needles and solid polymers[END_REF], this coefficient stands for radiant re-emission only and its value is maximized, providing h = 22W.m -2 .K -1 . Linearized values of this coefficient provide h in the range 18 -20W.m -2 .K -1 regarding the reference temperature (ignition or mean temperature between ambient and ignition temperature). The convective heat transfer coefficient can also be added to this coefficient, as suggested in [START_REF] Quintiere | Fundamentals of Fire Phenomena[END_REF], however natural convection on a flat surface is often expected to be negligible regarding radiant re-emission. This assumption is questionable for small targets according to [START_REF] Long | Scale and Transport Considerations on piloted Ignition of PMMA[END_REF]; however, in the case of thermally thick material, it seems acceptable. For thermally thin materials, the total heat transfer coefficient should then be a bit higher. Finally, in [START_REF] Grishin | A Mathematical Modelling of Forest Fires and New Methods of Fighting Them[END_REF], this coefficient is set at 10W.m -2 .K -1 . Thus, a value in the range 10 -20W.m -2 .K -1 is used in this study, which seems an acceptable range. Therefore, considering wood, τ KH ∈ [270; 4000]s; For polymer, τ KH ∈ [260; 10000]s while τ KH ∈ [2; 150]s for insulating foams. It must be noticed that [START_REF] Mowrer | An analysis of effective thermal properties of thermally thick materials[END_REF] suggests slightly lower values in the case of very low thermal inertia. These ranges are summarized in table 1, showing (for wood and polymer) the relevancy of series expansion for ignition times ranging from a few seconds to a few hundred seconds. For insulating foam, the efficiency of series expansion observed in [START_REF] Reska | A methodology for the estimation of ignition delay times in forest fire modelling[END_REF] in case of ignition time up to 400s however suggests that the efficient characteristic time τ KH for foam is much greater than table 1 values. Hence, practical dimensionless ignition time t * (defined as t * = t ig /τ KH ) for thermally thick solids will be considered lower or equal to one.

For thermally thin solid targets, such as solid particles composing a forest fuel layer, the solution suggested in [START_REF] Quintiere | Fundamentals of Fire Phenomena[END_REF] is used, providing the thermally thin kernel k(t):

k(t) = 1 h 1 -exp - hσ ρCp t . ( 5
)
where σ is the specific area of the solid particles. The thermally thin kernel characteristic time is thus τ k = ρCp/hσ. Since the ratio τ k /τ KH = h/λσ for a given material stands for its convective Biot number, it seems acceptable for the characteristic time τ k to be considered small in comparison with τ KH . Thanks to the expression of the convective Biot number, the latter can be estimated for typical forest fuel layer particle, considering σ in the range 600 -12000m -1 using values from [18,[START_REF] Cheney | The influence of fuel, weather and fire shape variables on fire-spread in grasslands[END_REF][START_REF] Catchpole | Rate of spread of free-burning fires in woody fuels in a wind tunnel[END_REF]. It provides τ k /τ KH ∈ [0.005; 0.555] for woody fuel particles. However, the thermally thin limit provided by [START_REF] Benkoussas | Modelling thermal degradation of woody fuel particles[END_REF] suggests that some of these particles can behave as thermally thick (for radiative Biot number Bi > 0.1), which corresponds to σ < 2640m -1 for a 10kW.m -2 radiant heat flux (10kW.m -2 seems to be an acceptable approximation for the heat flux lower limit, according to critical heat fluxes measured in [START_REF] Simeoni | Flammability studies for wildland and wildland-urban interface fires applied to pine needles and solid polymers[END_REF]). Hence, the specific area will be considered to range approximately from 3000m -1 to 12000m -1 for particles behaving as thermally thin. Thus, the kernel characteristic time τ k is in the range 1.35 -20s for the thinner woody fuel particles and in the range 30 -444s for the thicker woody fuel particles that still behave as thermally thin.

Temperature response to radiant heat fluxes viewed as Fourier series

The following assumption is now suggested: Effects of radiative heating variations and convective transfer variations can both be represented by apparent incoming heat flux variations. However, to mimic convective heat transfer variations, the apparent incoming heat flux variation magnitude increases as the radiative Biot number decreases, since the thermally thinner particles are the most sensitive to convective heat transfer modifications. Consequently, effects of flame unsteadiness on the radiant heat flux, practical uncontrolled variations of heat loss (convective and radiative) and periodic flaming contact (responsible for both incoming heat flux enhancement and heat loss decrease) could be modeled using periodic incoming heat flux variations. This is why the temperature response to a Fourier decomposed heat flux is investigated, providing thanks to Eq.( 2), using the heat conduction equation linearity, for an arbitrary kernel K 0 and a radiant heat flux Φ(t) = Φ 0 + (a n cos(ωt) + b n sin(ωt)):

θ(t) = Φ 0 K 0 (t ig ) + ω n I c - n I s , ( 6
)
where ω = 2πn/t ig ,

I c n = b n tig t=0 K 0 (t ig -τ ) cos(ωτ )dτ , (7)
and

I s n = a n tig t=0 K 0 (t ig -τ ) sin(ωτ )dτ . ( 8
)
Integrals I c n and I s n are solved analytically for the different kernels suggested in this study yet expressions are not provided here for clarity.

In order to account for the different periodic and quasi-periodic heat flux fluctuations involved in realistic ignition situations, pulsation range of buoyant diffusion flames and pool fires are taken as a reference, using for instance results of [START_REF] Malalasekera | A review of reaserch and an experimental study on the pulsation of buoyant diffusion flames and pool fires[END_REF], based on the pioneer work of [START_REF] Pagni | Pool fire vortex shedding frequencies[END_REF]. Hence, considering equivalent fire diameters ranging from 10 -2 m to 5m to account for pine needle litter fires up to crown fires, the reference frequencies are expected to be in the range 0.84 -16.8Hz. It corresponds to period in the range 0.06 -1.33s. This range provides a first insight on these frequencies, however higher and lower frequencies should also be considered since other physical phenomenon such as wind gusts, wind/fire front interactions and wind/vegetation interactions could be responsible for a larger frequency range. For instance the study of [START_REF] Morandini | Fire spread experiment across Mediterranean shrub: Influence of wind on flame front properties[END_REF] suggests heat flux variations due to wind gusts, with frequencies in the range 0.0006 -0.1Hz.

A criteria to estimate ignition time uncertainty

The solution θ(t) is now studied for a radiant heat flux Φ(t) = Φ 0 + b n sin(ωt). In order to investigate the effect of the harmonic part and thus ignition time uncertainty, the following expression f is firstly suggested, accounting for the ratio of the temperature rise due to the harmonic part of the radiant heat flux and the temperature rise due to the constant part:

f (K 0 , n) = ωI c n Φ 0 K 0 (t ig ) . ( 9
)
An harmonic dephasing, which is obviously a parameter that is not controlled in a practical ignition case, should also be taken into account. This is why f possible variation range ∆f = 2|f | will be considered to account for ignition time uncertainty. This function is calculated for the different kernels and then studied regarding the harmonic mode n, the perturbation relative magnitude φ = b n /Φ 0 (with 0 < φ < 1) and the dimensionless time to ignition t * = t ig /τ K0 . As suggested earlier, for thermally thick solids, φ is mostly accounting for radiant heat flux variations. Since experimental radiant heat fluxes from fire fronts in [START_REF] Chetehouna | On the experimental determination of flame front positions and of propagation parameters for a fire[END_REF] exhibits only small magnitude temporal variations, φ is practically expected to be small. On the contrary, for thermally thin solids, φ can also account for heat loss variations which suggests that φ can increase almost up to 1 when mimicking flame contact on thermally thin particles.

Thermally thick target

In the case of thermally thick target with high thermal inertia, function f is depending only on n and φ:

f (K, n) = - 1 2 φ S Fresnel (2 √ n) √ n , ( 10
)
where S Fresnel is the sinusoidal Fresnel integral. For n ≥ 1, the global trend exhibits a relatively slow yet always decreasing rate regarding n as shown on figure 1 (the curve t * = 0 corresponds to the high thermal inertia case).

For thermally thick target with low thermal inertia, function f is expressed the following way:

f (K H , n) = f (K + K h , n) = f (K, n) + f (K h , n) , (11)
f (K h , n) = 1 2 φ t * nπ - 1 4 φ t * C Fresnel (2 √ n) n 3/2 π , ( 12
)
where C Fresnel is the cosinusoidal Fresnel integral. As previously mentioned, t * is practically expected to be lower or equal to one. Consequently, the first term in f (K h , n) is always lowering the effect of the term f (K, n) and f (K H , n) global trend is practically monotonic as shown on figure 1, where ∆f (K H , n) is plotted for different values of t * in the range t * ∈ [0; 1]. Effects of the apparent incoming heat flux harmonic part on the ignition are lowered by surface losses which are damping the system response. This last remark allows the physical interpretation of figure 1 to be done regardless of the thermal inertia. Hence, according to Eq.( 10), frequencies that could be responsible for large ignition time uncertainties are independent of the kernel characteristic time. They rather depend on the ignition time. For instance, in the case of high thermal inertia, setting n > 7 leads to ∆f /φ < 0.2, which means that the potential effect of any harmonic mode n > 7 on the temperature rise leading to ignition is lower than 20% of the harmonic perturbation relative magnitude. Setting a 50% perturbation relative magnitude, if this perturbation oscillates more than seven times before theoretical ignition (i.e. ignition if no perturbation occurs), the solid surface temperature rise error due to harmonic perturbations at theoretical ignition time is lower than 10% of the temperature rise needed to achieve ignition, what can be considered as acceptable for fire safety application. Relevant frequencies are even lower for low thermal inertia: For instance, when t * = 0.5, ∆f /φ < 0.2 for n > 5.

Considering the previous example with a 50% perturbation relative magnitude, radiation variation due to flame puffing could then be neglected, even for the larger flame fronts considered in this study, as long as t ig ≥ 10s. It can be neglected for even smaller ignition time in the case of low thermal inertia materials. For lower frequency phenomenon such as wind gusts, uncertainty can be estimated comparing the considered physical phenomenon period with the ignition time and using Eq.( 10), Eq.( 11) and Eq.( 12).

Thermally thin target

For thermally thin target, function f is expressed the following way: This relation shows that a non-monotonic behavior of f can occur for n ≥ 1. Indeed, the kernel and the heat flux solicitation can resonate since df (k, n)/dn = 0 when n res = t * /2π, which corresponds to T res = 2πτ k where T res is the resonance period. Hence ∆f max = φ when n = n res , showing clearly that some harmonic variations of the heat flux (with period in the range 8.5-250s for thin forest fuel particles) can greatly modify the ignition time of thermally thin targets. A plot of f (k, n)/φ is provided on figure 2 for t * = 10, t * = 40 and t * = 100 to show the width of the resonating band.

f (k, n) = -φ 2πnt * 4π 2 n 2 + t * 2 . ( 13
) n [-] ∆f/φ [-] 10
Therefore, in the case of thin forest fuel particles, flame puffing effects of the larger fire fronts considered in this study are responsible for ∆f /φ = 0.3. This is 50% higher than in the thermal thick case when t ig ∼ 10s. This value is however constant which mean that ∆f /φ = 0.3 even when t ig → ∞. Hence, the relative uncertainty on t ig is constant but the absolute uncertainty grows as the ignition time increases. In the most general case (which means considering not only the finest fuel elements and considering any fire size), flame puffing frequencies and also any higher frequencies (for instance due to wind/vegetation interactions) are not expected to interact with the kernel and thus will not generate uncertainty on the ignition time calculation. However, lower frequencies due to wind gusts or wind/fire front interactions could be responsible for great uncertainties on the ignition time calculation if theses frequencies are in the range 0.005 -0.15Hz, regardless of the ignition time.

Concluding remarks

Using the classical ignition theory and a Fourier decomposition for the incoming heat flux on different kind of solid targets, this study quantifies how harmonic fluctuations of the heat flux can affect ignition times regarding a main constant contribution. Hence, for thermally thick target, only slow time-varying fluctuations can affect ignition times due to the characteristic time scale of the associated kernel. Frequencies which have relevant contributions on the ignition time uncertainty are depending linearly on the inverse of the ignition time. However, for a thermally thin target, the characteristic time scale of the kernel allows the later to resonate with the solicitation, showing a higher sensitivity (in comparison with the thermally thick case) to any frequency, plus the existence of particular frequencies (based on the kernel characteristic time) which interact actively with the ignition process. These frequencies are depending solely on the kernel characteristic time, contrary to the thermally thick case where it was depending on the ignition time. Hence, thermally thin kernel frequencies can be compared with the characteristic frequency of periodic (or quasi-periodic) physical phenomenon involved in fire propagation such as flame puffing, wind gusts, wind/fire front interactions and wind/vegetation interactions, showing that lowest frequency phenomenon such as wind gusts are the most likely to generate large uncertainties on the ignition time calculation.

Figure 1 .

 1 Figure 1. ∆f (K H , n)/φ for t * = 0, t * = 0.25, t * = 0.5 and t * = 1

Figure 2 .

 2 Figure 2. ∆f (k, n)/φ for t * = 10, t * = 40 and t * = 100

Email address: aymeric.lamorlette@univ-amu.fr (Aymeric Lamorlette).