Philippe Lebacque
email: philippe.lebacque@univ-fcomte.fr

Alexey Zykin
email: alzykin@gmail.com

ON M-FUNCTIONS ASSOCIATED WITH MODULAR FORMS

Keywords: 2010 Mathematics Subject Classification. Primary 11F11, Secondary 11M41 L-function, cuspidal newforms, value-distribution, density function

Let f be a primitive cusp form of weight k and level N, let χ be a Dirichlet character of conductor coprime with N, and let L(f ⊗ χ, s) denote either log L(f ⊗ χ, s) or (L ′ /L)(f ⊗ χ, s). In this article we study the distribution of the values of L when either χ or f vary. First, for a quasi-character ψ : C → C × we find the limit for the average Avg χ ψ(L(f ⊗ χ, s)), when f is fixed and χ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of L(f ⊗ χ, s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avg h f ψ(L(f, s)), when f runs through the set of primitive cusp forms of given weight k and level N → ∞. Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for L(f ⊗ χ, s).

1. Introduction 1.1. Some history. The study of the distribution of values of L-functions is a classical topic in number theory. In the first half of 20th century Bohr, Jessen, Wintner, etc. intiated a study of the distribution of the values of the logarithm log ζ(s) and the logarithmic derivative (ζ ′ /ζ)(s) of the Riemann zeta-function, when Re s = σ > 1 2 is fixed and Im s = τ ∈ R varies ( [START_REF] Bohr | Uber die Werteverteilung der Riemannschen Zetafunktion, I[END_REF], [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF], [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF], [START_REF] Kershner | On the asymptotic distribution of ζ ′ /ζ(s) in the critical strip[END_REF]). This was later generalized to L-functions of cusp forms and Dedekind zeta-functions by Matsumoto ([23], [START_REF] Matsumoto | Value-distribution of zeta-functions[END_REF], [START_REF] Matsumoto | Asymptotic probability measures of zeta-functions of algebraic number fields[END_REF]).

In the last decade Y. Ihara in [START_REF] Ihara | On "M -functions" closely related to the distribution of L ′ /L-values[END_REF] proposed a novel view on the problem by studying other families of L-functions. His initial motivation was to investigate the properties of the Euler-Kronecker constant γ K of a global field K, which was defined by him in [START_REF] Ihara | On the Euler-Kronecker constants of global fields and primes with small norms, Algebraic Geometry and Number Theory[END_REF] to be the constant term of the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function of K, ζ ′ K (s)/ζ K (s). The study of L ′ (1, χ)/L(1, χ) initiated in [START_REF] Ihara | On the logarithmic derivatives of Dirichlet L-functions at s = 1[END_REF] grew out to give a whole range of beautiful results on the value distribution of L ′ /L and log L.

Given a global field K, i.e. a finite extension of Q or of F q (t), and a family of characters χ of K Ihara considered in [START_REF] Ihara | On "M -functions" closely related to the distribution of L ′ /L-values[END_REF] the distribution of L ′ (s, χ)/L(s, χ) in the following cases:

(A) K is Q, a quadratic extension of Q or a function field over F q , χ are Dirichlet characters on K; (B) K is a number field with at least two archimedean primes, and χ are normalized unramified Grössencharacters; (C) K = Q and χ = χ t , t ∈ R defined by χ t (p) = p -it . The equidistribution results of the type

Avg ′ χ Φ L ′ (s, χ) L(s, χ) = C M σ (w)Φ(w)|dw|, (1)
(with a suitably defined average in each of the above cases) were proven for σ = Re s > 1 for number fields, and for σ > 3/4 for function fields, under significant restrictions on the test function Φ. The function field case was treated once again in [START_REF] Ihara | On L-functions over function fields: power-means of error-terms and distribution of L ′ /L-values, Algebraic Number Theory and Related Topics[END_REF] by Y. Ihara and K. Matsumoto, with both the assumptions on Φ and on σ having been relaxed (Φ of at most polynomial growth and σ > 1/2 respectively). The most general results in the direction of the case (A) were established in [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF] conditionally under the Generalized Riemann Hypothesis (GRH) in the number field case and unconditionally in the function field case (the Weil's Riemann hypothesis being valid) for both families L ′ (s, χ)/L(s, χ) and log L(s, χ). For Re s > 1 2 Ihara and Matsumoto prove that

Avg χ Φ L ′ (s, χ) L(s, χ) = C M σ (w)Φ(w)|dw|, Avg χ Φ(log L(s, χ)) = C M σ (w)Φ(w)|dw|,
for continuous test functions Φ of at most exponential growth. Note that Avg ′ in (1) is different from the one used in the latter paper, since extra averaging over conductors is assumed in the former case, the resulting statements being weaker.

Unconditional results for a more restrictive class of Φ (bounded continous functions), and with extra averaging over the conductor Avg ′ , but still for Re s > 1 2 were established in [START_REF] Ihara | On certain mean values and the value-distribution of logarithms of Dirichlet L-functions[END_REF] and [START_REF] Ihara | On the value-distribution of logarithmic derivatives of Dirichlet L-functions, Analytic number theory, approximation theory, and special functions[END_REF] in the log and log ′ cases respectively in the situations (A, K = Q) and (C).

The above results give rise to the density functions M σ (z) and a related function Ms (z 1 , z 2 ) (which is the inverse Fourier transform of M σ , when z 2 = z1 , s = σ ∈ R) both in the log and log ′ cases. Under optimal circumstances (though it is very far from being known unconditionally in all cases) we have

M σ (z) = Avg χ δ z (L(χ, s)) , Mσ (z 1 , z 2 ) = Avg χ ψ z 1 ,z 2 (L(χ, s)) ,
where L(s, χ) is either L ′ (s, χ)/L(s, χ) or log L(s, χ), δ z is the Dirac delta function, and ψ z 1 ,z 2 (w) = exp i 2 (z 1 w + z 2 w) is a quasi-character. The functions M and M turn out to have some remarkable properties that can be established unconditionally. For example, M has an Euler product expansion, an analytic continuation to the left of Re s > 1/2, its zeroes and the "Plancherel volume" C | Mσ (z, z)| 2 |dz| are interesting objects to investigate. We refer to [START_REF] Ihara | On certain arithmetic functions M (s; z 1 , z 2 ) associated with global fields: analytic properties[END_REF], [START_REF] Ihara | An analytic function in 3 variables related to the value-distribution of log L, and the "Plancherel volume[END_REF] for an in-depth study of M and M , as well as to the survey [START_REF] Ihara | On the value-distribution of log L and L ′ /L, New directions in valuedistribution theory of zeta and L-functions[END_REF] for a thorough discussion of the above topics.

In a recent paper by M. Mourtada and K. Murty [START_REF] Mourtada | Distribution of values of L ′ /L(σ, χ D )[END_REF] averages over quadratic characters were considered. Using the methods from [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF], they establish an equidistribution result conditional on GRH. Note that in their case the values taken by the L-functions are real. In this respect the situation is similar to the one considered by us in §5 in case we assume that s is real.

Finally, let us quote a still more recent preprint by K. Matsumoto and Y. Umegaki [START_REF] Matsumoto | On the value-distribution of the difference between logarithms of two symmetric power L-functions[END_REF] that treats similar questions for differences of logarithms of two symmetric power L-functions under the assumption of the GRH. Their approach is based on [START_REF] Ihara | On certain mean values and the value-distribution of logarithms of Dirichlet L-functions[END_REF] rather than on [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF], though the employed techniques are remarkably close to the ones we apply in §5. The results of Matsumoto and Umegaki are complementary to ours, since the case of Sym 1 f = f, which is the main subject of our paper, could not be treated in [START_REF] Matsumoto | On the value-distribution of the difference between logarithms of two symmetric power L-functions[END_REF].

1.2. Main results. In this article, we generalize to the case of modular forms the methods of Ihara and Matsumoto to understand the average values of L-functions of Dirichlet characters over global fields.

Our results are obtained in two different setting. First, we consider the case of a fixed modular form, while averaging with respect to its twists by Dirichlet characters. Our results in this setting are fairly complete, though sometimes conditional on GRH. Second, we consider averages with respect to primitive forms of given weight and level, when the level goes to infinity.

Let us formulate our main results. A more thorough presentation of the corresponding notation can found in §2 and in the corresponding sections.

Let B k (N) denote the set of primitive cusp forms of weight k and level N, let f ∈ B k (N), and let χ be a Dirichlet character of conductor m coprime with N. Define L(f ⊗ χ, s) to be either

(L ′ /L)(f ⊗ χ, s) or log L(f ⊗ χ, s), put g(f ⊗ χ, s, z) = exp iz 2 L(f, s) .
We introduce l z (n) to be the coefficients of the Dirichlet series expansion g(f ⊗χ, s, z) = n≥1 l z (n)n -s . Using the relations between the coefficients of the Dirichlet series expansion L(f, s)

= n≥1 η f (n)n -s , one can write l z (n) = x≥1 c N z,x (n)η f (x), where c N z,x (n) depend only on the level N. Put c z,x (n) = c 1 z,x (n).
In what follows, the expressions of the form f ≪ g, g ≫ f, and f = O(g) all denote that |f | ≤ c|g|, where c is a positive constant. The dependence of the constant on additional parameters will be explicitly indicated (in the form ≪ ǫ,δ,... or O ǫ,δ,... ), if it is not stipulated otherwise in the text. We denote by v p (n) the p-adic valuation of n, writing as well p k n if v p (n) = k. We also use the notation := or =: meaning that the corresponding object to the left or to the right of the equality respectively is defined in this way.

Our main results are as follows.

Theorem (Theorem 3.1). Assume that m is a prime number and let Γ m denote the group of Dirichlet characters modulo m. Let 0 < ǫ < 1 2 and T, R > 0. Let s = σ + it belong to the domain σ ≥ ǫ + 1 2 , |t| ≤ T, let z and z ′ be inside the disk D R = {z | |z| ≤ R}. Then, assuming the Generalized Riemann Hypothesis (GRH) for L(f ⊗ χ, s), we have

lim m→∞ 1 |Γ m | χ∈Γm g(f ⊗ χ, s, z)g(f ⊗ χ, s, z ′ ) = n≥1 l z (n)l z ′ (n)n -2σ =: Mσ (-z, z ′ ).
Theorem (Theorem 4.1). Let Re s = σ > 1 2 and let m run over prime numbers. Let Φ be either a continuous function on C with at most exponential growth, or the characteristic function of a bounded subset of C or of a complement of a bounded subset of C. Define M σ as the inverse Fourier transform of Mσ (z, z). Then under GRH for L(f ⊗ χ, s) we have

lim m→∞ 1 |Γ m | χ∈Γm Φ(L(f ⊗ χ, s)) = C M σ (w)Φ(w)|dw|.
Theorem (Theorem 5.1). Assume that N is a prime number and that k is fixed. Let 0 < ǫ < 1 2 and T, R > 0. Let s = σ + it belong to the domain σ ≥ ǫ + 1 2 , |t| ≤ T, and z and z ′ to the disc D R of radius R. Then, assuming GRH for L(f, s), we have

lim N →+∞ f ∈B k (N ) ω(f )g(f, s, z)g(f, s, z ′ ) = n,m∈N n -s m -s x≥1 c z,x (n)c z ′ ,x (m),
where ω(f ) are the harmonic weights defined in §5.

Finally, let us describe the structure of the paper. In §2 we introduce the notation and some technical lemmas to be used throughout the paper. The §3 is devoted to the proof of Theorem 3.1 on the mean values of the logarithms and logarithmic derivatives of L-functions obtained by taking averages over the twists of a given primitive modular form. Using GRH, we deduce it from Ihara and Matsumoto's results. In §4 we study unconditionally the analytic properties of M and M functions in the above setting. We then prove an equidistribution result (Theorem 4.1), which is, once again, conditional on GRH. In §5, we consider the average over primitive forms of given weight k and level N, when N → ∞, establishing under GRH Theorem 5.1. The orthogonality of characters is replaced by the Petersson formula in this case, which obviously makes the proofs trickier. Finally, open questions, remarks and further research directions are discussed in §6.

Acknowledgements.

We would like to thank Yasutaka Ihara for helpful discussions. The first author would like to express his gratitude to the INRIA team GRACE for an inspirational atmosphere accompanying his stay, during which a large part of this work was done.

Notation

The goal of this section is to introduce the notation necessary to state our main results. We also prove some auxiliary estimates to be used throughout the paper.

2.1. The g-functions. Let N, k be two integers. We denote by S k (N) the set of cusp forms of weight k and level N, and by S new k (N) the set of new forms. For f ∈ S k (N) we write

f (z) = ∞ n=1 η f (n)n (k-1)
/2 e(nz) for its Fourier expansion at the cusp ∞, with the standard notation e(nz) = e 2πinz .

Let B k (N) denote the set of primitive forms of weight k and level N, i.e. the set of f nor = f /η f (1) where f runs through an orthogonal basis of S new k (N) consisting of eigenvectors of all Hecke operators T n , so that the Fourier coefficients of the elements of B k (N) are the same as their Hecke eigenvalues. Note that for a primitive form f ∈ B k (N) all its coefficients η f (n) are real.

The L-function of a primitive form f ∈ B k (N) is defined as the Dirichlet series

L(f, s) = ∞ n=0 η f (n)n -s .
The series converges absolutely for Re s > 1, however, L(f, s) can be analytically continued to an entire function on C. It admits the Euler product expansion:

L(f, s) = p L p (f, s),
where, for any prime number p,

L p (f, s) = (1 -η f (p)p -s + p -2s ) -1 if (p, N) = 1, (1 -η f (p)p -s ) -1 if p | N.
By the results of Deligne, these local factors can be written as follows [START_REF] Iwaniec | Topics in classical automorphic forms[END_REF]Chapter 6] or [20, Chapter IX, §7]):

L p (f, s) = 1 -α f (p)p -s -1 1 -β f (p)p -s -1 , (2)
where

     |α f (p)| = 1, β f (p) = α f (p) -1 if (p, N) = 1, α f (p) = ±p -1 2 , β f (p) = 0 if p N (that is p | N and p 2 ∤ N), α f (p) = β f (p) = 0 if p 2 | N.
We are interested in the two functions

       g(f, s, z) = exp iz 2 L ′ (f, s) L(f, s) , G(f, s, z) = exp iz 2 log L(f, s) .
Define h n (x) and H n (x) as the coefficients of the following generating functions:

exp xt 1 -t = +∞ n=0 h n (x)t n , exp(-x log(1 -t)) = +∞ n=0 H n (x)t n ,
or, equivalently (cf. [11, §1.2]), as the functions given by h 0 (x) = H 0 (x) = 1 and, for n ≥ 1,

h n (x) = n r=0 1 r! n -1 r -1 x r , H n (x) = 1 n! x(x + 1) . . . (x + n -1).
As we have iz 2

L ′ (f, s) L(f, s) = - iz 2 p α f (p)p -s log p 1 -α f (p)p -s + β f (p)p -s log p 1 -β f (p)p -s ,
we can write (using the standard convention that, in the case when β f (p) = 0, we put β f (p) n = 0, if n > 0, and β f (p) 0 = 1):

g(f, s, z) = exp iz 2 L ′ (f, s) L(f, s) = = p exp α f (p)p -s 1 -α f (p)p -s • -iz log p 2 exp β f (p)p -s 1 -β f (p)p -s • -iz log p 2 = = p n h n - iz 2 log p α f (p) n p -ns n h n - iz 2 log p β f (p) n p -ns = = p +∞ n=0 n r=0 h r - iz 2 log p h n-r - iz 2 log p α f (p) r β f (p) n-r p -ns = = p∤N +∞ n=0 n r=0 h r - iz 2 log p h n-r - iz 2 log p α f (p) 2r-n p -ns • • p N +∞ n=0 h n - iz 2 log p α f (p) n p -ns =: p +∞ n=0 λ z (p n )p -ns .
In a similar way we get:

G(f, s, z) = exp iz 2 log L(f, s) = = p exp - iz 2 log(1 -α p (f )p -s ) exp - iz 2 log(1 -β p (f )p -s ) = = p∤N +∞ n=0 n r=0 H r iz 2 H n-r iz 2 α f (p) 2r-n p -ns p N +∞ n=0 H n iz 2 α f (p) n p -ns =: p +∞ n=0 Λ z (p n )p -ns .
We extend multiplicatively λ z and Λ z to N so that we can write:

g(f, s, z) = n≥1 λ z (n)n -s , G(f, s, z) = n≥1 Λ z (n)n -s .
We will use the notation L for L ′ (f, s) L(f, s) or log L(f, s), g for g or G, h z (p n ) for h n -iz 2 log p or H n iz 2 , and l for λ or Λ depending on the case we consider. Thus, we can write in a uniform way:

g(f, s, z) = exp iz 2 L(f, s) = n≥1 l z (n)n -s = p +∞ n=0 l z (p n )p -ns = = p∤N +∞ n=0 n r=0 h z (p r )h z (p n-r )α f (p) 2r-n p -ns p N +∞ n=0 h z (p n )α f (p) n p -ns .
The coefficients l z (n) will be used to define the M -functions in the case of averages over twists of modular forms by Dirichlet characters.

2.2.

The coefficients l z (n) and c z,x (n). In this subsection we will find a more explicit expression for l z (n). For p ∤ N we will use the formula (see [30, (3.5)])

η f (p r ) = α f (p) r+1 -β f (p) r+1 α f (p) -β f (p) ,
which easily follows from [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF]. Taking into account that β f (p) = ᾱf (p), we have for r ≥ 2

η f (p r ) = α f (p) r+1 -α f (p) r+1 α f (p) -α f (p) = r i=0 α f (p) i α f (p) r-i = r i=0 α f (p) r-2i = = α f (p) r + α f (p) r + r-1 i=1 α f (p) r-2i = α f (p) r + α f (p) r + r-2 i=0 α f (p) r-2i-2 = = α f (p) r + α f (p) r + η f (p r-2 ).
The above formula also holds for r = 1 if we put η f (p -1 ) = 0. From this we deduce that

α f (p) r + β f (p) r = η f (p r ) -η f (p r-2 ).
Using the previous formula, we can write

l z (p r ) = r a=0 h z (p a )h z (p r-a )α f (p) 2a-r = h z (p r 2 ) 2 + ⌊ r-1 2 ⌋ a=0 h z (p a )h z (p r-a ) α f (p) r-2a + α f (p) 2a-r = h z (p r 2 ) 2 + ⌊ r-1 2 ⌋ a=0 h z (p a )h z (p r-a ) η f (p r-2a ) -η f (p r-2a-2 ) = h z (p r 2 ) 2 -h z (p r 2 -1 )h z (p r 2 +1 ) + ⌊ r-1 2 ⌋ a=0 (h z (p a )h z (p r-a ) -h z (p a-1 )h z (p r-a+1 ))η f (p r-2a ) = ⌊ r 2 ⌋ a=0 (h z (p a )h z (p r-a ) -h z (p a-1 )h z (p r-a+1 ))η f (p r-2a ),
where we put h z (p

r 2 ) = h z (p r 2 -1 ) = 0, if r is odd, and h z (p a ) = 0, if a < 0. When p | N we have l z (p r ) = h z (p r )α f (p) r = h z (p r )η f (p) r = h z (p r )η f (p r ).
Denoting by P the set of prime numbers, for n = p∈P p vp(n) put

I N (n) = {m ∈ N | v p (m) ≡ v p (n) mod 2 for p ∈ P, v p (n) = v p (m) if p | N} and J N (n) = {m ∈ I N (n) | v p (m) ≤ v p (n) for all p ∈ P}.
Note the following easy estimate ([3, Theorem 315]) in which τ (n) is the number of divisors of n:

|J N (n)| = p|n v p (n) 2 + 1 ≤ τ (n) ≪ ǫ n ǫ . (3)
The previous computations may be summarized as follows:

l z (p r ) = x∈J N (p r ) c N z,x (p r )η f (x),
where

c N z,p a (p r ) =      h z (p r-a 2 )h z (p r+a 2 ) -h z (p r-a 2 -1 )h z (p r+a 2 +1 ), if p ∤ N and r ≡ a mod 2, h z (p r ), if p | N and r = a, 0, otherwise.
We have

l z (n) = p|n l z (p vp(n) ) and η f (n)η f (m) = η f (nm) if (n, m) = 1, thus l z (n) = p|n   x∈J N (p vp(n) ) c N z,x (p vp(n) )η f (x)   = x∈J N (n) c N z,x (n)η f (x), with c N z,x (n) = p|n c N z,p vp(x) (p vp(n) ).
Note that the coefficients c N z,x (n), I N (n), and J N (n) depend only on the level N and not directly on the modular form f. Let us also define

I(n) = I 1 (n), J(n) = J 1 (n), and c z,x (n) = c 1 z,x (
|c N z,x (p r )| ≤ e 2 √ r-a 2 |z| log p e 2 √ r+a 2 |z| log p + e 2 √ ( r-a 2 -1)|z| log p e 2 √ ( r+a 2 +1)|z| log p ≤ e 2 √ |z| log p √ r-a 2 + √ r+a 2 + e 2 √ |z| log p √ r-a 2 -1+ √ r+a 2 +1 ≤ e 2 √ |z| log p √ 2r + e 2 √ |z| log p √ 2r ≤ 2e 2 √ 2r|z| log p .
when p ∤ N. The above estimates on h z (p r ) also imply the same bound on c N z,x (p r ) when p | N. Now, denoting by ω(n) the number of distinct prime divisors of n and using once again the concavity of

√ x, for n = p∈P p vp(n) we have log |c N z,x (n)| ≤ p|n (log 2 + v p (n) log p √ 8R) ≪ R   p|n v p (n) log p   √ 8R ≪ R p|n v p (n) log p ω(n) ≪ R log n 2 + log log n log n,
since by [5, Sublemma 3.10.5] (which is classical in the case of N) we have

ω(n) ≪ log n 2 + log log n . (4)
We thus conclude that |c N z,x (n)| ≪ ǫ,R n ǫ . As for the second statement, we notice that the estimate (3) together with Deligne bound

|η f (n)| ≤ τ (n) ≪ ǫ n ǫ imply l z (n) ≪ ǫ |J N (n)| • n ǫ • τ (n) ≪ ǫ n 3ǫ .
We conclude the section by the following trivial but useful lemma.

Lemma 2.2. We have l z (n) = l -z (n), and c N z,x (n) = c N -z,x (n).
Proof. The eigenvalues η f (n) are all real, so the L-functions L(f, s) have Dirichlet series with real coefficients. Thus the statement of the lemma follows from the definition of the coefficients l z (n), and c N z,x (n).

Average on twists

This section is devoted to the proof of an averaging result for twists of a given primitive form. It is to a large extent based on the work of Ihara and Matsumoto [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF], which provides a general setting for the problem we consider.

3.1.

Setting. Let us fix a primitive cusp form f ∈ B k (N) of weight k and level N. Let χ : (Z/mZ) × → C × be a primitive character mod m, where (m, N) = 1. It is known (see [START_REF] Iwaniec | Analytic number theory[END_REF]Prop. 14.19 and Prop. 14.20]) that f ⊗ χ is a primitive form of weight k, level Nm 2 , and nebentypus χ 2 . We consider the twisted L-function given by

L(f ⊗ χ, s) = p L p (f ⊗ χ, s),
where the local factors are defined as follows:

L p (f ⊗ χ, s) = 1 -α f (p)χ(p)p -s -1 1 -β f (p)χ(p)p -s -1 ,
with the notation of §2. It is an L-function of degree 2 and conductor Nm 2 , entire and polynomially bounded in vertical strips. After multiplication by the gamma factor

γ k (s) = √ π2 3-k 2 (2π) -s Γ s + k -1 2 ,
it satisfies a functional equation [15, §5.11]. Its analytic conductor q(f ⊗ χ, s) is defined as follows:

q(f ⊗ χ, s) = Nm 2 s + k -1 2 + 3 s + k + 1 2 + 3 ≤ Nm 2 (|s| + k + 3) 2 .
Just as in §2 we use the following notation for modular forms with nebentypus:

       g(f ⊗ χ, s, z) = exp iz 2 L ′ (f ⊗ χ, s) L(f ⊗ χ, s) , G(f ⊗ χ, s, z) = exp iz 2 log L(f ⊗ χ, s) .
We also write g(f ⊗ χ, s, z) to denote either of the above two functions.

If G is a function on a finite group K, let Avg χ∈K G(χ) denote the usual average |K| -1 χ∈K G(χ).

3.2.

The M -function. We would like to understand the average over all Dirichlet characters mod m of the functions g(f ⊗ χ, s, z), when m runs through large prime numbers. Ihara and Matsumoto's results apply in this case and we get the following theorem.

Theorem 3.1. Assume that m is a prime number. Let Γ m denote the group of Dirichlet characters modulo m. Let 0 < ǫ < 1 2 and T, R > 0. Let s = σ + it belong to the domain σ ≥ ǫ + 1
2 , |t| ≤ T, let z and z ′ be inside the disk D R . Then, assuming the Generalized Riemann Hypothesis (GRH) for L(f ⊗ χ, s), in the notation of §2 we have

Avg χ∈Γm g(f ⊗ χ, s, z)g(f ⊗ χ, s, z ′ ) - (n,m)=1 l z (n)l z ′ (n)n -2σ ≪ ǫ,R,T,f m -ǫ 2 . ( 5
)
Moreover,

lim m→∞ Avg χ∈Γm g(f ⊗ χ, s, z)g(f ⊗ χ, s, z ′ ) = n≥1 l z (n)l z ′ (n)n -2σ . Proof. We notice that g(f ⊗ χ, s, z) = n≥1 l z (n)χ(n)n -s
, where l z (n) are the coefficients of g(f, s, z). We thus can deduce the theorem from [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF]Theorem 1]. We can pass to the situation treated in [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF] by omitting the summand corresponding to the trivial character χ 0 since in our case all the g(f ⊗ χ, s, z) are holomorphic for Re s > 1 2 . Thus, it is enough to prove that the family l |z|≤R is uniformly admissible in the sense of Ihara and Matsumoto. First of all, the property (A1), asserting that l |z|≤R (n) ≪ ǫ n ǫ , follows from Lemma 2.1.

The property (A2) states that g(f ⊗ χ, s, z) extend to holomorphic functions on Re s > 1 2

for any non trivial χ, which is true under GRH.

The property (A3) will be proven in the following lemma that will be used again in §5.

Lemma 3.2. Let f be a primitive form of weight N, and let χ be a primitive Dirichlet character of conductor m coprime with N. Then, assuming GRH for L(f ⊗ χ, s), we have for Re s

≥ 1 2 + ǫ : max(0, log |g(f ⊗ χ, s, z)|) ≪ ǫ,R ℓ(t) 1-2ǫ ℓ(mNk) 1-2ǫ , where ℓ(x) = log(|x| + 2), t = Im s.
Proof of the Lemma. First, the following estimates hold ([15, Theorems 5.17 and 5.19]) for any s with 1 2 < Re s = σ ≤ 5 4 :

- L ′ (f ⊗ χ, s) L(f ⊗ χ, s) = O 1 2σ -1 (log q(f ⊗ χ, s)) 2-2σ + log log q(f ⊗ χ, s) , and log L(f ⊗ χ, s) = O (log q(f ⊗ χ, s)) 2-2σ (2σ -1) log log q(f ⊗ χ, s) + log log q(f ⊗ χ, s) ,
the implied constants being absolute.

Next, for the same range of s we have log q(f ⊗ χ, s) ≪ log(mNk) + log(|t| + 2) ≪ ℓ(mNk) + ℓ(t).

Thus we see that

log |g(f ⊗ χ, s, z)| = log exp iz 2 L(f ⊗ χ, s) = Re iz 2 L(f ⊗ χ, s) ≪ R |L(f ⊗ χ, s)|, so max(0, log |g(f ⊗ χ, s, z)|) ≪ ǫ,R ℓ(t) 1-2ǫ ℓ(Nmk) 1-2ǫ . If σ ≥ 5
4 a much simpler estimate suffices. Indeed, using the fact that [15, (5.25)]

- L ′ (f, s) L(f, s) = n Λ f (n) n s and log L(f, s) = - n Λ f (n) n s log n , with Λ f (n) supported on prime powers and Λ f (p n ) = (α f (p) n + β f (p) n ) log p, we see that both L ′ (f, s) L(f, s)
and log L(f ⊗ χ, s) are bounded by an absolute constant. Thus the conclusion of the lemma still holds in this case.

Thus Ihara and Matsumoto's property (A3) is established (with a stronger bound than required), since in our case N and k are fixed. So, the family we consider is indeed uniformly admissible.

Remark 3.3. The estimate [START_REF] Ihara | On "M -functions" closely related to the distribution of L ′ /L-values[END_REF] should still be true if we omit the condition on m to be prime. To prove it one establishes an analogue of Lemma 3.2, replacing χ with the primitive character by which it is induced and estimating the bad factors of the L-function (with some additional work required when m is not coprime with N). Then one uses once again [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF]Theorem 1], in which the first inequality is true without any restriction on the conductor.

Remark 3.4. The theorem should hold unconditionally for σ = Re s > 1 by orthogonality of characters, all the series being absolutely convergent in this domain.

As a direct consequence, we obtain the following result on averages of the values of g. Put

Ms (z 1 , z 2 ) = ∞ n=1 l z 1 (n)l z 2 (n)n -2s .
Because of Lemma 2.1, the series converges uniformly and absolutely on Re

s ≥ 1 2 + ε, |z 1 |, |z 2 | ≤ R, defining a holomorphic function of s, z 1 , z 2 for Re s > 1 2 . Put ψ z 1 ,z 2 (w) = exp i 2 (z 1 w + z 2 w) .
Corollary 3.5. Let m run over prime numbers. Then, assuming GRH,

lim m→∞ Avg χ∈Γm ψ z 1 ,z 2 (L(f ⊗ χ, s)) = Mσ (z 1 , z 2 ).
Proof. By definition, we have :

ψ z 1 ,z 2 (L(f ⊗ χ, s)) = exp i 2 z 1 L(f ⊗ χ, s) exp i 2 z 2 L(f ⊗ χ, s) = g(f ⊗ χ, s, -z 1 )g(f ⊗ χ, s, z 2 ).
By Theorem 3.1 we get

lim m→∞ Avg χ∈Γm ψ z 1 ,z 2 (L(f ⊗ χ, s)) = n≥1 l -z 1 (n)l z 2 (n)n -2σ .
Lemma 2.2 implies that l -z (n) = l z (n), so the corollary is proven.

The distribution of L-values for twists

Our next result concerns the distribution of the values of logarithmic derivatives and logarithms of L-functions of twists of a fixed modular form f . In this section the dependence on f in ≪ will be omitted.

Recall that we have defined

Ms (z 1 , z 2 ) = ∞ n=1 l z 1 (n)l z 2 (n)n -2s ,
the corresponding series being absolutely and uniformly convergent on Re s

≥ 1 2 + ǫ, |z 1 | ≤ R, |z 2 | ≤ R. For σ ∈ R, we put Mσ (z) = Mσ (z, z).
Define the family of additive characters

ψ z 1 ,z 2 (w) = exp i 2 (z 1 w + z 2 w) .
We also let ψ z (w) = ψ z,z (w) = exp(i Re(z w)). Recall that the Fourier transform of φ :

C → C, φ ∈ L 1 is defined as F φ(z) = C φ(w)ψ z (w)|dw| = 1 2π C φ(w)e i Re(z w) |dw| = 1 2π R 2 φ(w)e i(xx ′ +yy ′ ) dxdy, where |dw| = 1 2π dxdy, x = Re w, y = Im w, x ′ = Re z, y ′ = Im z.
The goal is to prove the following equidistribution result, which is an analogue of [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF]Theorem 4].

Theorem 4.1. Let Re s = σ > 1
2 and let m run over prime numbers. Let Φ be either a continuous function on C with at most exponential growth, that is Φ(w) ≪ e a|w| for some a > 0, or the characteristic function of a bounded subset of C or of a complement of a bounded subset of C. Define M σ as the inverse Fourier transform of Mσ (z), M σ (z) = F Mσ (-z). Then under GRH for L(f ⊗ χ, s) we have

lim m→∞ Avg χ∈Γm Φ(L(f ⊗ χ, s)) = C M σ (w)Φ(w)|dw|. ( 6
)
Remark 4.2. The above theorem should hold unconditionally for any σ > 1 and any continuous function Φ on C, by virtue of Remarks 3.4 and (iv) of Corollary 4.12.

To prove this theorem we first construct the local M and M -functions and establish their properties. We then obtain a convergence result for partial M-functions M s,P for finite sets of primes P to a global function M. This allows us to prove some crucial estimates for the growth of M. Finally, we deduce the global result using corollary 3.5. Our approach is strongly influenced by that of Ihara and Matsumoto, the main ingredients being inspired by the results of Jessen and Wintner [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF] that we have to adapt to our situation.

All the results below, except from the proof of Theorem 4.1 itself, do not depend on GRH.

4.1. The functions M s,P and Ms,P . Let Re s = σ > 0. Define the functions on

T p = C 1 = {t ∈ C | |t| = 1} by g s,p (t) = -(log p)α(p)p -s t 1 -α(p)p -s t + -(log p)β(p)p -s t 1 -β(p)p -s t , and G s,p (t) = -log(1 -α(p)p -s t) -log(1 -β(p)p -s t).
As before, we let g s,p denote either g s,p or G s,p , depending on the case we consider. We note that the local factor of the L-function is 1 once p 2 | N, so we can omit such primes from our considerations.

Denote by f p (z) the expression

-(log p)α(p)z 1 -α(p)z + -(log p)β(p)z 1 -β(p)z or -log(1 -α(p)z) -log(1 -β(p)z).
in the log ′ and log case respectively. Note that if p ∤ N,

f p (z) = -log p • η f (p)z -2z 2 1 -η f (p)z + z 2 or -log(1 -η f (p)z + z 2 ) respectively. The functions f p (z) are holomorphic in the open disc |z| < 1. We obviously have g s,p (t) = f p (p -s t).
For a prime number p, let T p = C 1 be equipped with the normalized Haar measure

d × t = dt 2πit
. If P is a finite set of primes, we let T P = p∈P T p and we denote by d × t P the normalized Haar measure on T P . Put also g s,P = p∈P g s,p .

We introduce the local factors Ms,p (z 1 , z 2 ) via

Ms,p (z 1 , z 2 ) = +∞ r=0 l z 1 (p r )l z 2 (p r )p -2rs . ( 7
)
The series is absolutely and uniformly convergent on compacts in Re s > 0 by Lemma 2.1. Put Ms,P (z 1 , z 2 ) = p∈P Ms,p (z 1 , z 2 ). We also define Mσ,p (z) = Mσ,p (z, z), and Mσ,P (z) = Mσ,P (z, z).

Lemma 4.3. (i) The function Ms,P (z 1 , z 2 ) is entire in z 1 , z 2 . (ii) We have Ms,p (z 1 , z 2 ) = C 1 exp i 2 (z 1 g s,p (t -1 ) + z 2 g s,p (t)) d × t.
In particular,

Mσ,p (z 1 , z 2 ) = C 1 ψ z 1 ,z 2 (g σ,p (t))d × t, and Mσ,p (z)
= C 1 exp(i Re(g σ,p (t)z))d × t.
(iii) The "trivial" bound | Mσ,p (z)| ≤ 1 holds.

Proof. (i) This is a direct corollary of the absolute and uniform convergence of the series of analytic functions [START_REF] Ihara | An analytic function in 3 variables related to the value-distribution of log L, and the "Plancherel volume[END_REF], defining Ms,p (z 1 , z 2 ).

(ii) It is clear from the definitions that exp iz 2 g s,p (t) = ∞ r=0 l z (p r )(p -s t) r . So, the statement is implied by the fact that Ms,p is the constant term of the Fourier series expansion of exp i 2 (z 1 g s,p (t -1 ) + z 2 g s,p (t)) . (iii) Obviously follows from (ii).

For the sake of convenience in what follows we will identify a function on R 2 with the Radon measure or the tempered distribution it defines, when the latter make sense. We will also regard the Fourier transform or the convolution products as being defined via the corresponding distributions. We refer to [18, §2, §3] for more details. (iv) Let P be a set of primes. If |P ∩P f | > 4, then M σ,P admits a continuous density (still denoted by M σ,P ) which is an L 1 function. The function M σ,P satisfies M σ,P (z

) = M σ,P (z) ≥ 0. (v) M σ,P is of class C r once |P ∩ P f | > 2(r + 2).
Proof. (i) The uniqueness statement is obvious and the existence is given by the direct image measure (g s,P ) * (d × t P ). The volume of an open set U of R 2 is thus given by M σ,P (U) = Vol(g -1 s,P (U)), therefore M σ,P has compact support equal to the image of g s,P and mass 1. From the formula M s,P (Φ) = T P Φ(g s,P (t P ))d × t P , it is clear that M s,P depends only on σ, since Haar measures on T P are invariant under multiplication by p i Im(s) .

(ii) From the definition of the convolution product we note that, regarded as distributions with compact support, M σ,P = * p∈P M σ,p .

Next, F M σ,P = F ( * p∈P M σ,P ) = p∈P F M σ,p . From Lemma 4.3 we see that Mσ,P (z 1 , z 2 ) = M σ,P (ψ z 1 ,z 2 ), and for the Fourier transforms of tempered distributions on C ≃ R 2 we have

F M σ,p (φ) = M σ,p C ψ z (w)φ(w)|dw| = Tp C ψ gs,p(t) (w)φ(w)|dw|d × t = C Tp ψ gs,p(t) (w)φ(w)d × t|dw| = C M σ,p (ψ z (w))φ(w)|dw| = C M σ,p (ψ w (z))φ(w)|dw| = C Mσ,p (w)φ(w)|dw|.
We deduce that F M σ,P = Mσ,P (z).

(iii) This is the most delicate part. Unfortunately, we cannot apply Jessen-Wintner theorem [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 13] to f p (z), since ρ 0 (in the notation of the latter theorem) depends on p. Therefore, we need to establish the following explicit version of their result.

ρ ′′′ = |a 1 | √ 2 k≥2 k 3 |a k |ρ k-2
, for any r < ρ 0 = min(ρ, ρ ′′′ ) we have

Dr (z) ≪ r,F (1 + |z|) -1 2 .
Proof. Our goal is to make the proof of [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 13] explicit in order to be able to estimate ρ 0 . To do so, we will verify the conditions of [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 12] by proceeding in several steps.

First of all, we want to ensure that F ′ (z) = 0, and the curve S is Jordan. Put

ρ ′ = |a 1 | √ 2 k≥2 k|a k |ρ k-2 .
If r < min(ρ, ρ ′ ), we have F ′ (z) = 0 for all z ∈ D r = B(0, r), and F is injective on D r . Indeed,

either | Re a 1 | or | Im a 1 | is greater than |a 1 | √ 2 .
Without loss of generality we can suppose that

| Re a 1 | ≥ |a 1 | √ 2 . Then | Re F ′ (z)| ≥ | Re a 1 | -|z| k≥2 k|a k |ρ k-2 ≥ |a 1 | √ 2 -|z| k≥2 k|a k |ρ k-2 > 0 on D r , in particular F ′ (z) = 0.
The sign of Re F ′ (z) does not change as the function is continuous, so once more, without loss of generality, we may assume that Re F ′ (z) > 0.

Then, for z 1 = z 2 two points in D r , we have by convexity of D r ,

Re F (z 2 ) -F (z 1 ) z 2 -z 1 = 1 0 Re F ′ (z 1 + t(z 2 -z 1 ))dt > 0,
which proves the injectivity. Thus F is a conformal transformation and S is a Jordan curve. The next step is to get a condition for the curve S to be convex. We use a well-known criterion [29, Part 3, Chapter 3, 108], stating that S is convex if

Re zF ′′ (z) F ′ (z) > -1 on |z| = r. The estimate Re zF ′′ (z) F ′ (z) ≤ |zF ′′ (z)| |F ′ (z)| ≤ |z| k≥2 k(k -1)|a k |ρ k-2 |a 1 | -|z| k≥2 k|a k |ρ k-2 ≤ |z| k≥2 k(k -1)|a k |ρ k-2 |a 1 | 1 -1 √ 2
for r < min(ρ, ρ ′ ) implies that the condition is satisfied once the left-hand side is less than one, that is [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 12] is satisfied for all r < ρ. As for (ii) we consider the function

r < ρ ′′ = |a 1 |(2 - √ 2) 2 k≥2 k(k -1)|a k |ρ k-2 . Now, the condition (i) of
g τ (θ) = k≥1 |a k |r k cos 2π(kθ + γ k -τ ),
where τ ∈ [0, 1) is fixed and a k = |a k |e 2πiγ k . We have to prove that for r explicitly small enough, its second derivative has exactly two roots on [0, 1). We compute

h τ (θ) = - g ′′ τ (θ) 4π 2 r = |a 1 | cos 2π(θ + γ 1 -τ ) + r k≥2 j 2 |a k |r k-2 cos 2π(kθ + γ k -τ ), so h ′ τ (θ) = -2π|a 1 | sin 2π(θ + γ 1 -τ ) -2πr k≥2 k 3 |a k |r k-2 sin 2π(kθ + γ k -τ ). Take now r < |a 1 | √ 2 k≥2 k 3 |a k |ρ k-2 = ρ ′′′ . Since k≥2 k 3 |a k |ρ k-2 ≥ k≥2 k 2 |a k |ρ k-2
, the function h τ can possibly have zeroes only on the two intervals (modulo 1) containing ± 1 4 -γ 1 + τ mod 1 defined by the condition

| cos 2π(θ + γ 1 -τ )| < 1 √ 2 .
The same argument shows that h τ is positive at θ = -γ 1 + τ mod 1 and negative at θ = 1 2 + τ -γ 1 mod 1, and therefore it has at least one zero in each of these intervals.

On the other hand, when

| cos 2π(θ + γ 1 -τ )| < 1 √ 2 , we see that |h ′ τ (θ)| ≥ 2π|a 1 || sin 2π(θ + γ 1 -τ )| -2πr k≥2 k 3 |a k |r k-2 > 2π|a 1 | 1 - 1 2 - 1 √ 2 = 0,
showing that there is exactly one zero of h τ in each of the above intervals.

We thus can apply [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 12], obtaining that the conclusion of the theorem holds for r < ρ 0 = min(ρ, ρ ′ , ρ ′′ , ρ ′′′ ) = min(ρ, ρ ′′′ ). By [28, Corollary 2 of Theorem 4], there exists a set P of positive density such that, for all p ∈ P, |η f (p)| > 1. We apply the above lemma to the functions F = f p , p ∈ P, defined by absolutely convergent series for |z| < ρ + ǫ, with ρ = ǫ = 1 2 , and to the radii r p = p -σ . In the log case, the coefficient |a 1 | of the lemma is |η f (p)|, whereas we have for any i, |a i | ≤ 2. In the log ′ case, the coefficients are all multiplied by log p : |a 1 | is |η f (p)| log p and |a i | ≤ 2 log p. Thus, for p such that p ∈ P and

p -σ < 1 8 √ 2 k≥2 k 3 2 -k = 1 204 √ 2 ,
we have that Mσ,p (z

) = O (1 + |z|) -1 2
, proving thus (iii).

(iv), (v) By the Fourier inversion formula, we get F Mσ,P (-z) = M σ,P . It is well-known [18, §3] that f = F g is absolutely continuous and admits continuous density, once the integral C |g(w)||dw| converges. Moreover, it possesses continuous partial derivatives of order ≤ p, if the convergence holds for C |z| p |g(w)||dw|. Thus, to deduce the regularity properties of M σ,P it suffices to bound the growth of Mσ,P (z).

For the primes p / ∈ P f , we use the trivial bound | Mσ,p (z)| ≤ 1 from Lemma 4.3. For all the other p the bound from (iii) can be applied. Now, the identity M σ,P (z) = M σ,P (z) is the consequence of (ii) together with the symmetry Mσ (z, z) = Mσ (z, z). The positivity of M σ,P (z) follows from the definition M σ,P (U) = Vol(g -1 σ,P (U)) together with the continuity that we have established. Let P denote the set of all prime numbers, P x = {p ∈ P | p ≤ x}. Proof. This comes directly from the fact that P f has positive density, implying that there exists C, such that if x ≥ C, then (P x \ P y ) ∩ P f contains more than 2r + 4 primes.

Remark 4.7. The previous proposition is motivated by the following equidistribution result that is essentially implied by [START_REF] Ihara | On "M -functions" closely related to the distribution of L ′ /L-values[END_REF]Lemma 4.3.1] applied to Ψ = Φ • g σ,P :

lim m→∞ Avg χ∈Γm Φ(L P (f ⊗ χ, s)) = T P Φ(g σ,P (t P ))d × t P ,
where Φ is a an arbitrary continuous function on C, L P is either the logarithm or the logarithmic derivative of the corresponding partial product p∈P L p (f ⊗ χ, s) for L(f ⊗ χ, s), and χ runs through all Dirichlet character of prime conductor m ∈ P. Note a difference in the type of average considered in the aforementioned lemma with the one we use. The proof stays the same, being an application of Weyl's equidistribution criterion together with the orthogonality of characters.

Note, however, that it is not at all obvious to pass from the local equidistribution result to the global one. This also seems to give (after very significant effort) only a certain weaker form of global averaging results (e.g. [START_REF] Ihara | On "M -functions" closely related to the distribution of L ′ /L-values[END_REF], [START_REF] Ihara | On L-functions over function fields: power-means of error-terms and distribution of L ′ /L-values, Algebraic Number Theory and Related Topics[END_REF]). Following later papers by Ihara and Matsumoto, we use instead the convergence for particular test functions (quasi-characters, c.f. Theorem 3.1) and then deduce the general case, using the information on the resulting distributions together with some general statements on convergence of measures.

4.2.

Global results for Mσ . Let us establish some global properties of M , in particular the convergence of Mσ,P to Mσ . From now on we assume that Re s = σ > 1 2 , without mentioning it in each statement. Proof. (i) This is a direct corollary of the absolute and uniform convergence of the series of analytic functions, defining Ms (z 1 , z 2 ).

(ii) To prove the uniform convergence of the infinite product it is enough to establish it for the sum

p | Ms,p (z 1 , z 2 ) -1|. By Lemma 2.1 we see that | Ms,p (z 1 , z 2 ) -1| ≤ ∞ r=1 |l z 1 (p r )||l z 2 (p r )|p -2rσ ≪ ǫ ′ ,R ∞ r=1 p (2ǫ ′ -2σ)r ≤ ∞ r=1 p (-1-ǫ)r < 2p -1-ǫ .
which implies the convergence.

The limit of the infinite product equals Ms . Indeed, the series for Ms converges absolutely and uniformly, thus the difference between Ms and the partial product over primes p ≤ x, which is

n∈Sx l z 1 (n)l z 2 (n)n -2s
, where S x is the set of integers n divisible by at least one prime strictly greater than x, tends to 0 as x → ∞.

(ii) Note that for any two sets P ⊂ P ′ of primes, any z ∈ C,

| Mσ,P ′ (z)| ≤ | Mσ,P (z)|.
Corollary 4.6 implies that one can find a finite set of primes P such that Mσ,P (z) ≪ (1 + |z|) -N . This is enough to conclude. Remark 4.9. Along the same lines as in [10, 3.20], one proves a more precise estimate: | Mσ,p (z) -1| ≪ |z| 2 p -2σ in the log case, and | Mσ,p (z) -1| ≪ |z| 2 p -2σ log p in the log ′ case with absolute constants in ≪ . Proof. First of all, the notation x ≫ 0 is used to make sure that all the elements of the sequence are continuous functions. Fix ǫ > 0. One can find a closed disk D r and x ′ large enough, so that for all

P ′ ⊃ P x ′ , C\Dr | Mσ,P ′ (w)||dw| < ǫ.
The sequence ( Mσ,Px (z) x≫0 converges uniformly to Mσ (z) = Mσ (z) on D r by Proposition 4.8, thus we can find x ′′ large enough to guarantee for x > max(x ′ , x ′′ ),

F Mσ (z) -F Mσ,Px (z) ∞ < 2ǫ.
This proves that (F Mσ,Px (z)

) x≫0 = (M σ,Px (-z)) x≫0 converges uniformly to F Mσ (z) = M σ (-z).
The same arguments apply if we remove P y from the set of all primes. Moreover, taking the Fourier transform of Mσ = Mσ,Py × p / ∈Py Mσ,p , we see that M σ (z) = M σ,Py * ( * P\Py M σ,p ). Corollary 4.12. We have

(i) M σ (z) = M σ (z) ≥ 0; (ii) C M σ (z)|dz| = 1;
(iii) M σ (z) ∈ C ∞ and the partial derivatives of M σ,Px converge uniformly to those of M σ ;

(iv) If σ > 1, the support of M σ is compact.

Proof. (i) This is obvious from the corresponding properties of M σ,P .

(ii) Using the identity M σ (z) = F Mσ (-z), we see that

C M σ (z)|dz| = Mσ (0) = 1.
(iii) We note that, given p, there exists y 0 such that for p > y 0 , M σ,Py has continuous partial derivatives up to order p. Now, letting (iv) Indeed, by the uniform convergence of M σ,P to M σ , and the fact that the support of M σ,P is equal to the image of g s,P , it is enough to prove that the latter is bounded for σ > 1. This is true since the series p p -σ converges for σ > 1.

D (a,b) = ∂ a+b ∂ a z ∂ b z , we have D (a,b) (f * g) = (D (a,b) f ) * g, if
We will now obtain the rapid decay of M σ à la Jessen-Wintner by proving the following proposition, which is crucial for the proof of the main theorem of this section. Proposition 4.13. For any λ > 0, M σ (z) = O σ,λ (e -λ|z| 2 ), as |z| → ∞. The same is true for all its partial derivatives.

Proof. We adapt the proof of Jessen-Wintner [START_REF] Jessen | Distribution functions and the Riemann Zeta Functions[END_REF]Theorem 16] to our specific case. The proof is based on an argument of Paley and Zygmund.

Let σ > 1 2 and λ > 0 be fixed. Let p 1 < • • • < p i . . . denote the sequence of all prime numbers. Write P j = {p 1 , . . . , p j }. We have

f p (z) = i≥1 a i,p z i , on the disk B(0, 1). By writing 1 -η f (p)z + z 2 = (1 -α f (p)z)(1 -β f (p)z)
, where |α f (p)| and |β f (p)| are less than or equal to 1, we see that for all i, |a i,p | ≤ 2 log p in the log ′ case and ≤ 2 in the log case respectively.

Put r p = p -σ . Then the series p |a 1,p | 2 r 2 p converges, so that we can find q such that

d = 1 -2λ p>pq |a 1,p | 2 r 2 p > 0.
For n > q let us look at the partial sums

s n (θ 1 , . . . , θ n ) = n j=1
f p j (r p j e iθ j ) and t n (θ q+1 , . . . , θ n ) = n j=q+1 a 1,p j r p j e iθ j , where θ j ∈ [0, 2π]. We can bound the difference by

|s n (θ 1 , . . . , θ n ) -t n (θ q+1 , . . . , θ n )| ≤ q j=1 f p j (r p j e iθ j ) + n j=q+1 ∞ k=2 |a k,p j |r k p j ≤ q j=1 f p j (r p j e iθ j ) + 2 n j=q+1 r 2 p j (1 -r p j ) -1 log p j ≤ q j=1 f p j (r p j e iθ j ) + 8 +∞ j=q+1 r 2 p j log p j ≤ q j=1 sup ϑ j ∈[0,2π] |f p j (r p j e iϑ j )| + 8 +∞ j=q+1 r 2 p j log p j ≪ A(q) as (1 -r p j ) -1 ≤ √ 2 √ 2 -1 .
Here A depends only on q and not on n.

By an inequality of Jessen [17, p. 290-291], writing

|s n | 2 ≤ 2|s n -t n | 2 + 2|t n | 2 , we obtain T Pn exp(λ|s n (θ 1 , . . . , θ n )| 2 )dθ 1 . . . dθ n ≤ e 2λA(q) 2 T Pn,q exp(2λ|t n (θ q+1 , . . . , θ n )| 2 )dθ q+1 . . . dθ n , ≤ e 2λA(q) 2 1 -2λ n j=q+1 |a 1,p | 2 r 2 p j ≤ e 2λA(q) 2 d -1 = K. ( 8
)
where P n,q = P n \ P q . Noting that M σ,Pn (e λ|w| 2 ) is just the left-hand side of (8), we deduce:

M σ,Pn (e λ|w| 2 ) ≤ K,
where K is independent of n. Thus by Fatou lemma and Proposition 4.11 we conclude that

C M σ (w)e λ|w| 2 dw ≤ K.
Let us take y such that M σ,Py is a continuous function. It is clear that if we remove all the terms corresponding to p ≤ y, and take q > y large enough we obtain exactly the same bound for the function M Let ρ be large enough, so that D ρ contains the support of M σ,Py . Then

M σ (z) = (M σ,Py * M (y) σ )(z) = C M σ,Py (w)M (y) σ (z -w)|dw| = Dρ M σ,Py (w)M (y) σ (z -w)|dw| ≤ sup Dρ M σ,Py (w) • C M (y) σ (z -w)|dw| ≤ Ke -λ(|z|-ρ) 2 sup Dρ M σ,Py (w).
As y, ρ, ρ are independent of z, we obtain that

M σ (z) = O(e -λ|z| 2 ).
According to Corollary 4.6, one can take y large enough so that M σ,Py has continuous partial derivatives of order up to p. We also have

D (a,b) (f * g) = D (a,b) (f ) * g = f * D (a,b) (g).
Thus, the same arguments as above imply that the required estimate holds for partial derivatives of M σ (z) of any order p.

Corollary 4.14. The functions M σ (z) and Mσ (z) belong to the Schwartz space, that is they go to zero as |z| → ∞ faster than any inverse power of |z|, as do all their derivatives.

Proof. The statement is clear for M σ (z) by the above theorem. Now, Mσ (z) = F M σ (-z). Since F maps Schwartz functions to Schwartz functions the result follows.

Mσ (z 1 , z 2 ) = C M σ (w)ψ z 1 ,z 2 (w)|dw|.
Proof. Each side of the above equality is an entire function of z 1 , z 2 (the left one by Proposition 4.8, the right one by Proposition 4.13). These functions are equal when z 2 = z1 by Proposition 4.11, thus they must coincide for any z 1 , z 2 ∈ C.

Remark 4.16. The last corollary also follows from Theorem 4.1, however we prefer to give a direct proof. 4.4. Proof of Theorem 4.1. We will apply Lemma A from [11, §5], which is a general result that allows to deduce from the convergence of averages for a special class of functions Φ, the same fact for more general Φ.

First of all, Corollaries 4.12 and 4.14 imply that M σ is a good density function on R 2 in the sense of Ihara and Matsumoto, that is, it is non-negative, real valued, continuous, with integral over R 2 equal to 1, and such that both the function and its Fourier transform belong to L 1 ∩ L ∞ . By 3.5 the identity (6) holds for any additive character ψ z of C. Lemma A implies then that ( 6) is true for any bounded continuous Φ, for the characteristic function of any compact subset of R 2 or of the complement of such a subset. Now, let us take φ 0 (r) = exp(ar). Proposition 4.13 implies that C M σ (z)φ 0 (|z|)|dz| converges. The same reasoning as in [11, §5.3, Sublemma] allows us to see that Avg χ∈Γm exp(a|L(f ⊗ χ, s)|) ≪ 1. This concludes the proof of Theorem 4.1.

Average on primitive forms

While working with modular forms it is analytically more natural to consider harmonic averages instead of usual ones. One introduces the harmonic weight

ω(f ) = Γ(k -1) (4π) k-1 (f, f ) N , where (f, f ) N = Γ 0 (N )\H |f (z)| 2 y k dx dy y 2 is the Petersson scalar product, H = {z ∈ C | Im z > 0}. We denote by Avg h f ∈B k (N ) G(f ) the harmonic average Avg h f ∈B k (N ) G(f ) = f ∈B k (N ) ω(f )G(f ).
It can be proven [16, Corollary 2.10 for m = n = 1] that for squarefree N we have

f ∈B k (N ) ω(f ) = ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 , (9)
thus Avg

h f ∈B k (N )
is an average operator when ϕ(N ) N → 1.

One has the following interpretation of ω(f ) via the symmetric square L-functions [16, Lemma 2.5]:

ω(f ) = 2π 2 (k -1)NL(Sym 2 f, 1) . ( 10
)
Theorem 5.1. Assume that N is a prime number and that k is fixed. Let 0 < ǫ < 1 2 and T, R > 0. Let s = σ + it belong to the domain σ ≥ ǫ + 1 2 , |t| ≤ T, and z and z ′ to a disc D R . Then, assuming GRH for L(f, s), for any δ > 0 we have

Avg h f ∈B k (N ) (g(f, s, z)g(f, s, z ′ )) - n,m∈N n -s m -s x∈J(n)∩J(m) (nm,N )=1 c z,x (n)c z ′ ,x (m) ≪ ǫ,R,T,δ,k N -ǫ/2+δ ,
and lim

N →+∞ Avg h f ∈B k (N ) (g(f, s, z)g(f, s, z ′ )) = n,m∈N n -s m -s x∈J(n)∩J(m) c z,x (n)c z ′ ,x (m).
The convergence of the series is on the right-hand sides is uniform and absolute in the above domains without the assumption of GRH. Remark 5.2. In contrast to the situation, considered in Theorem 3.1, we see that the average depends both on Re s and Im s. In fact, the independence of Im s in the case of averages with respect to characters is the corollary of the invariance of Haar measures on C 1 under rotations.

Corollary 5.3. Under the conditions of the previous theorem we have

lim N →+∞ Avg h f ∈B k (N ) ψ z 1 ,z 2 (L(f, s)) = Mh s (z 1 , z 2 ) = n,m∈N n -s m -s x∈J(n)∩J(m) c z 1 ,x (n)c z 2 ,x (m)
.

Proof. We have

ψ z 1 ,z 2 (L(f, s)) = g(f, s, -z 1 )g(f, s, z 2 ), so lim N →+∞ Avg h f ∈B k (N ) ψ z 1 ,z 2 (L(f, s)) = n,m∈N n -s m -s x∈J(n)∩J(m) c -z 1 ,x (n)c z 2 ,x (m).
The corollary follows from the equality c -z,x (n) = c z,x (n) which is implied by Lemma 2.2.

Naive approach.

In this subsection we try to estimate the average in a naive way via Euler products. This approach works for Re s large enough and gives a formula which turns out to be valid for more general s. The intermediate calculations will be used again in §5.3.

All the estimates are written assuming only that N is squarefree and not assuming that k is fixed until the very end of §5.3.

We have

Avg h f ∈B k (N ) (g(f, s, z)g(f, s, z ′ )) = f ∈B k (N ) ω(f ) n,m≥1 n -s m -s l z (n)l z ′ (m). Let τ k (n) = |{(d 1 , . . . , d k ) ∈ N k | d 1 • • • • • d k = n}|.
We will use a version of the Petersson formula proven in [START_REF] Iwaniec | Low lying zeros of families of L-functions[END_REF]Corollary 2.10]. Note that our weights are slightly different from those used in [START_REF] Iwaniec | Low lying zeros of families of L-functions[END_REF], we follow instead [START_REF] Royer | Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique[END_REF] in our normalization.

Proposition 5.4. If N is squarefree, (m, N) = 1, (n, N 2 ) | N, then S(m, n) = f ∈B k (N ) ω(f )η f (m)η f (n) = ϕ(N) N δ(m, n) + ∆(m, n),
where δ(m, n) is the Kronecker symbol and

∆(m, n) = O k -5 6 (mn) 1 4 N -1 (n, N) -1/2 τ (N) 2 τ 3 ((m, n)) log(2mnN) ,
the implied constant being absolute.

The conditions of this proposition are in particular satisfied once (nm, N) = 1. We will also use the following trivial bound, when (m, N) = 1 :

|S(m, n)| ≤ f ∈B k (N ) ω(f ) τ (m)τ (n) (m, N) = ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 τ (m)τ (n) (m, N) , (11)
which holds by virtue of ( 9) and the fact that

|η f (m)| ≤ τ (m) (m, N) since N is squarefree.
Obviously, the corresponding bound is also true if we assume instead that (n, N) = 1.

Remark 5. 5. In what follows, one can possibly soften our restrictions on N (in particular, remove the assumption that N → ∞) by using more elaborate bounds on the sums in the case when (mn, N) = 1, applying directly the construction of an explicit basis of S k (N) from B k (N), in a way similar to [START_REF] Iwaniec | Low lying zeros of families of L-functions[END_REF]Proposition 2.6].

Using the above estimates, we can write

Avg h f ∈B k (N ) (g(f, s, z)g(f, s, z ′ )) = f ∈B k (N ) ω(f ) n,m n -s m -s l z (n)l z ′ (m) = n,m n -s m -s f ∈B k (N ) ω(f )l z (n)l z ′ (m) = n,m n -s m -s x∈J N (n),y∈J N (m) c N z,x (n)c N z ′ ,y (m) f ∈B k (N ) ω(f )η f (x)η f (y) = n,m n -s m -s x∈J N (n),y∈J N (m) (xy,N )=1 c N z,x (n)c N z ′ ,y (m) δ(x, y) ϕ(N) N + ∆(x, y) + n,m n -s m -s x∈J N (n),y∈J N (m) (xy,N ) =1 c N z,x (n)c N z ′ ,y (m) f ∈B k (N ) ω(f )η f (x)η f (y) = ϕ(N) N n,m n -s m -s x∈J N (n)∩J N (m) (x,N )=1 c N z,x (n)c N z ′ ,x (m) + n,m n -s m -s x∈J N (n),y∈J N (m) (xy,N )=1 c N z,x (n)c N z ′ ,y (m)∆(x, y) + n,m n -s m -s x∈J N (n),y∈J N (m) (xy,N ) =1 c N z,x (n)c N z ′ ,y (m)S(x, y).
The fact that the sum can be subdivided into three parts will be justified by the absolute convergence of the series for Re s large enough.

Put M (s) = n,m n -s m -s x∈J N (n)∩J N (m) (x,N )=1 c N z,x (n)c N z ′ ,x (
n -s m -s x∈J(n)∩J(m) (nm,N )=1 c z,x (n)c z ′ ,x (m).
Our goal is to verify that M (s) gives the principal term of the asymptotic behaviour of m(s, z, z ′ ). If m / ∈ I(n), which is equivalent to

I(m) = I(n), the term c z,x (n)c z ′ ,x (m) vanishes. Therefore, M(s) = n∈N, (nm,N )=1 m∈I(n) n -s m -s x∈J(n)∩J(m) c z,x (n)c z ′ ,x (m).
Let us define r -(n) to be the largest square dividing n, and r + (n) to be the least square divisible by n. So, if n = p k 1 1 . . . p k l l , we have

n = p k 1 mod 2 1 . . . p k l mod 2 l r -(n) 2 , r + (n) 2 = p k 1 mod 2 1 . . . p k l mod 2 l n,
and the squarefree part of n is equal to

p k 1 mod 2 1 . . . p k l mod 2 l = r + (n) r -(n) .
Using this notation, we can write for s = σ + it

M (s) = n≥1 r≥1 n -σ+it r + (n) r -(n) -σ-it r -2s x∈J(n)∩J(m) (mn,N )=1 c z,x (n)c z ′ ,x r + (n)r 2 r -(n) = n,r≥1 r + (n) -2σ r -(n) 2it r -2s x∈J(n) (mn,N )=1 c z,x (n)c z ′ ,x r + (n)r 2 r -(n) , (12)
so

| M(s)| ≤ n,r≥1 (r + (n)) -2σ r -2σ x∈J(n) (mn,N )=1 |c z,x (n)| c z ′ ,x r + (n)r 2 r -(n) .
There are 2 ω(n) -1 = 2 l -1 different n giving the same r + (n). As ω(n) ≪ log n 2+log log n by ( 4), so 2 l ≪ ǫ n ǫ , using Lemma 2.1 and (3) we see that the the sum M (s) converges absolutely for Re s > 1/2 : (mn) -σ+ 1 4 +ǫ .

| M(s)| ≪ ǫ n,r≥1 n ǫ • n -2σ • r -2σ • n ǫ • n ǫ • r -2ǫ • n ǫ = n≥1 n -2σ+4ǫ
In a similar way, putting

∆ ′ (s) = n,m≥1 n -s m -s x∈J N (n),y∈J N (m) (xy,N ) =1 c N z,x (n)c N z ′ ,y (m)S(x, y),
we get

|∆ ′ (s)| ≪ ǫ 1 p min (N) ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 m,n≥1
(mn) -σ+ǫ , where p min (N) is the least prime factor of N. These bounds only make sense for σ = Re s > 5/4, when the series converge. For these values of s we conclude that the error terms tend to 0, once p min (N) → ∞ (recall that we assume N to be squarefree). In the next section we are going to show how the estimates can be pushed to the left from Re s > 5/4.

∈ B k (N), 0 < ǫ ′ < ǫ, T > 0, for | Im(s)| ≤ T we have |g -(f, s, z, X)| ≪ ǫ ′ (NkX) ǫ ′ X -ǫ .
Proof. Once again our proof largely mimics that of [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF]Proposition 2.2.13]. We need to estimate the integral

g -(f, s, z, X) = 1 2πi Re w=ǫ ′ -ǫ Γ(w)g(f, s + w, z)X w dw.
Clearly, |X w | = X ǫ ′ -ǫ and it is well-known [11, (2.2.9)] that

Γ(w) ≪ | Im w| c-1/2 exp - π 2 | Im(w)| ,
when | Im w| ≥ 1, Re w ≤ c, so in our case Γ(w) ≪ exp(-| Im(w)|). Lemma 3.2 ensures that, putting u = Im(w) and t = Im(s), we have

log |g(f, s + w, z)| ≪ ℓ(kN) 1-2ǫ ′ ℓ(t + u) 1-2ǫ ′ .
Therefore, there exists

C = C(T, ǫ ′ ) such that |g(f, s + w)| ≤ exp Cℓ(Nk) 1-2ǫ ′ (log(|u| + 1)) 1-2ǫ ′ ≤ exp Cℓ(Nk) 1-2ǫ ′ log(|u| + 1) .
So, by comparison with the Γ-integral, we have

|g -(f, s + w, z, X)| ≪ X ǫ ′ -ǫ +∞ 0 e -u (u + 1) Cℓ(N k) 1-2ǫ ′ du ≪ X ǫ ′ -ǫ Γ(Cℓ(Nk) 1-2ǫ ′ + 1) ≪ X ǫ ′ -ǫ exp(Cℓ(Nk) 1-2ǫ ′ log(Cℓ(Nk) 1-2ǫ ′ )) ≪ X ǫ ′ -ǫ exp(C ′ ℓ(Nk) 1-2ǫ ′ log(ℓ(Nk))) ≪ ǫ ′ X ǫ ′ -ǫ exp(ǫ ′ ℓ(Nk)) ≪ X ǫ ′ -ǫ (Nk) ǫ ′ ,
since for Nk large enough depending on T and ǫ ′ , C ′ ℓ(Nk) -2ǫ ′ log(ℓ(Nk)) < ǫ ′ holds.

5.3.

Averaging. We now go back to averaging over primitive forms. We denote for simplicity g = g(s, f, z), g ′ = g(s, f, z ′ ) and we adopt similar notation for g ± and g ′ ± . First of all, using the decomposition established in §5.2, we note that

Avg h f ∈B k (N ) (gg ′ ) = Avg h f ∈B k (N ) (g + g ′ + ) -Avg h f ∈B k (N ) (g + g ′ -) -Avg h f ∈B k (N ) (g -g ′ + ) + Avg h f ∈B k (N ) (g -g ′ -).
Our first goal is to prove that the average

Avg h f ∈B k (N ) (g + g ′ + ) = f ∈B k (N ) ω(f ) n,m≥1 n -s m -s l z (n)l z ′ (m)e -n+m X .
gives the main term of the asymptotic behaviour. The calculations of §5.1 allow us to decompose the above average as follows:

Avg h f ∈B k (N ) (g + g ′ + ) = M (s, X) + ∆(s, X) + ∆ ′ (s, X), (13) with
M(s, X) = ϕ(N) N n,m n -s m -s e -n+m X x∈J N (n)∩J N (m) (x,N )=1 c N z,x (n)c N z ′ ,x (m), ∆(s, X) = n,m n -s m -s e -n+m X x∈J N (n),y∈J N (m) (xy,N )=1 c N z,x (n)c N z ′ ,y (m)∆(x, y), ∆ ′ (s, X) = n,m n -s m -s e -n+m X x∈J N (n),y∈J N (m) (xy,N ) =1 c N z,x (n)c N z ′ ,y (
m)S(x, y).

Noting that 0 < 1 -e -a < min(a, 1) and fixing any α > 0, we see that

| M (s) -M (s, X)| = ϕ(N) N n,m n -s m -s (1 -e -n+m X ) x∈J N (n)∩J N (m) (x,N )=1 c N z,x (n)c N z ′ ,x (m) ≤ ϕ(N) N n≤αX m≤αX n -σ m -σ n + m X x∈J N (n)∩J N (m) (x,N )=1 |c N z,x (n)||c N z ′ ,x (m)| + ϕ(N) N n≥αX or m≥αX n -σ m -σ (1 -e -n+m X ) x∈J N (n)∩J N (m) (x,N )=1 |c N z,x (n)||c N z ′ ,x (m)|.
The calculations of §5.1 together with the observation that (in the notation of ( 12)) r 2 r + (b) 2 = mn result in the following bound valid for any ǫ ′′ > 0:

n≥αX or m≥αX n -σ m -σ (1 -e -n+m X ) x∈J N (n)∩J N (m) (x,N )=1 |c N z,x (n)||c N z ′ ,x (m)| ≪ ǫ ′′ (rs) 2 ≥αX (rs) -2σ+ǫ ′′ ≪ ǫ ′′ r≥ √ αX r -2σ+ǫ ′′ ≪ ǫ ′′ (αX) 1/2-σ+ǫ ′′ /2 ,
while the absolute convergence of the series for M(s) implies

n≤αX m≤αX n -σ m -σ n + m X x∈J N (n)∩J N (m) (x,N )=1 |c N z,x (n)||c N z ′ ,x (m)| ≪ α.
Taking ǫ ′′ small enough so that β = 1/2 -σ + ǫ ′′ /2 < 0 and α satisfying α = (αX) β , we finally see that

| M(s) -M (s, X)| ≪ ǫ ′′ ϕ(N) N X 1/2-σ+ǫ ′′ /2 1/2+σ-ǫ ′′ /2 ≤ ϕ(N) N X ǫ ′′ /2-ǫ . (14)
Now, let us turn to the second and the third terms in [START_REF] Ihara | On the logarithmic derivatives of Dirichlet L-functions at s = 1[END_REF]. Once again, applying the estimates from §5.1 we see that for any ǫ ′′ > 0

|∆(s, X)| ≪ ǫ ′′ τ (N) 2 log N Nk 5/6 m,n≥1 (mn) -σ+ 1 4 +ǫ ′′ e -m+n X , |∆ ′ (s, X)| ≪ ǫ ′′ 1 p min (N) ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 m,n≥1
(mn) -σ+ǫ ′′ e -m+n X .

Bounding the sums via the corresponding improper integrals (cf. [11, proof of Proposition 2.2.13]), we get

|∆(s, X)| ≪ ǫ ′′ τ (N) 2 log N Nk 5/6 X 3/2+2ǫ ′′ -2ǫ , ( 15
) |∆ ′ (s, X)| ≪ ǫ ′′ 1 p min (N) ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 X 1+2ǫ ′′ -2ǫ . (16)
In what follows, we will choose X (as a function of N) in such a way that the right-hand sides in ( 14), [START_REF] Iwaniec | Analytic number theory[END_REF], and ( 16) tend to 0. With this choice of X, taking z = z ′ and using the absolute convergence of M(s), we obtain

Avg h f ∈B k (N ) |g + | 2 ≪ ǫ ′′ 1, Avg h f ∈B k (N ) |g ′ + | 2 ≪ ǫ ′′ 1
Let us estimate the remaining terms involving g -and g ′ -. By Lemma 5.7 and ( 9)

Avg h f ∈B k (N ) |g -| 2 ≪ ǫ ′ ,T (NkX) 2ǫ ′ X -2ǫ f ∈B k (N ) ω(f ) ≤ ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 (NkX) 2ǫ ′ X -2ǫ .
We apply the Cauchy-Schwartz to get

| Avg h f ∈B k (N ) (g + g ′ -)| + | Avg h f ∈B k (N ) (g -g ′ + )| + | Avg h f ∈B k (N ) (g -g ′ -)| ≪ ǫ ′ ϕ(N) N + O τ (N) 2 log(2N) Nk 5/6 (NkX) 2ǫ ′ X -ǫ , (17) since
(NkX) 2ǫ ′ ≥ (NkX) ǫ ′ and X -2ǫ ≤ X -ǫ .
Let us now turn to the case considered in the theorem, by assuming that k is fixed and N = p is prime. Assuming that ǫ ′′ < 2ǫ, we have

| M(s) -M (s, X)| ≪ ǫ ′′ X ǫ ′′ /2-ǫ , |∆(s, X)| ≪ ǫ ′′ log p p X 3/2+2ǫ ′′ -2ǫ , |∆ ′ (s, X)| ≪ ǫ ′′ 1 √ p X 1+2ǫ ′′ -2ǫ , | Avg h f ∈B k (N ) (g + g ′ -)| + | Avg h f ∈B k (N ) (g -g ′ + )| + | Avg h f ∈B k (N ) (g -g ′ -)| ≪ ǫ ′ ,k p 2ǫ ′ X 2ǫ ′ -ǫ .
Taking X = p 1/2 , we see that the above bounds lead to

| Avg h f ∈B k (N ) g(f, s, z)g(f, s, z ′ ) -M (s)| ≪ δ p -ǫ/2+δ ,
where δ, which depends on ǫ ′ and ǫ ′′ , can be taken arbitrarily small. The second part of the theorem follows from the first.

Open questions and remarks

This section is devoted to a series of questions and remarks to complement the results of the paper. We hope to address at least some of them in subsequent articles. We start by the topics discussed in §3. Question 6.1. Can Theorem 3.1 be proven in a greater generality? For example, one can consider L-functions of more general automorphic cusp forms and the average taken with respect to their twists by Hecke characters of imaginary quadratic number fields or algebraic function fields with a fixed place at infinity. As indicated in [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF], going beyond imaginary quadratic number fields seems to be tricky since it involves essentially new problems related to the presence of non-trivial units. One can also consider averages over quadratic characters in the spirit of [START_REF] Mourtada | Distribution of values of L ′ /L(σ, χ D )[END_REF] Question 6.2. What is an unconditional version of Theorem 4.1?

The unconditional results [10, Theorem 1] and [START_REF] Ihara | On the value-distribution of logarithmic derivatives of Dirichlet L-functions, Analytic number theory, approximation theory, and special functions[END_REF]Theorem 1.1] suggest that it should be possible to prove similar statements in our case. Question 6.3. Prove an analogue of Theorem 4.1 for modular forms in the other situations within the framework of the cases (A), (B), (C) discussed in the introduction. Some results in this direction were established by Mastsumoto in [START_REF] Matsumoto | A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms[END_REF] in the case (C), that is the equidistribution of L(f, σ + it), when σ is fixed and t ∈ R varies. It seems, however, that, even when considering averages of Dirichlet L-functions conditionally on GRH, this question has not been fully investigated, the most advanced results having been obtain only in the case (A). Question 6.4. Carry out a more in-depth study of the functions M and M .

In the case of Dirichlet characters this was done in [START_REF] Ihara | On certain arithmetic functions M (s; z 1 , z 2 ) associated with global fields: analytic properties[END_REF], [START_REF] Ihara | An analytic function in 3 variables related to the value-distribution of log L, and the "Plancherel volume[END_REF]. One should be able to write down an explicit power series expansion of Ms (z 1 , z 2 ) in the variables z 1 , z 2 , establish its analytic continuation, study its growth, its zeroes, etc.

We next switch to the case of averages with respect to primitive forms of §5, where the results are far less complete. Question 6.5. Can one obtain Theorem 5.1 with weaker assumptions on N? Can we let k tend to infinity, while N is fixed? Can we let k + N → ∞?

By following carefully the proof of Theorem 5.1, one can see that the limit statement is still true when N = 1 and k → ∞. Indeed, in this case ∆ ′ is not present and the parameter X cas be chosen to be equal to k 1/2 . This suggests that some greater generality should be possible. The idea would be to use better bounds on averages of the Fourier coefficients of cusp forms with indices not coprime with N, which should be possible by a careful treatement of an explicit basis of the space of old forms in the spirit of [START_REF] Iwaniec | Low lying zeros of families of L-functions[END_REF] Question 6.6. Prove an unconditional version of Theorem 5.1. Surprisingly enough, a crude reasoning with Euler products does not seem to work even for Re s > 1. An unconditional version for Re s > 1/2 will certainly be tricky to obtain even if one only considers characters ψ z as in [START_REF] Ihara | On certain mean values and the value-distribution of logarithms of Dirichlet L-functions[END_REF] and [START_REF] Ihara | On the value-distribution of logarithmic derivatives of Dirichlet L-functions, Analytic number theory, approximation theory, and special functions[END_REF]. Question 6.7. Is it possible to establish value distribution results in the case harmonic averages over the set of primitive forms?

The reason we could not carry out the study analogous to that of §4 is the absence of a local theory (at least in a straight-forward way). Indeed, the Ms do not seem to admit an Euler product in this case. One could hope to rely on the interpretation of ω(f ) via the symmetric square L-functions [START_REF] Ihara | On certain mean values and the value-distribution of logarithms of Dirichlet L-functions[END_REF], though there does not seem to be an easy way to do that. Question 6.8. Can one remove the harmonic weights in Theorem 5.1?

At least two approaches are available. The papers [START_REF] Iwaniec | Low lying zeros of families of L-functions[END_REF], [START_REF] Kowalski | The analytic rank of J 0 (q) and zeros of automorphic L-functions[END_REF], [START_REF] Kowalski | A lower bound for the rank of J 0 (q)[END_REF] address a similar issue in different situations by using the interpretation (10) of the weights via L(Sym 2 f, 1).

A more conceptual way would be to construct the local theory first. The results of Serre [START_REF] Serre | Répartition asymptotique des valeurs propres de l'opérateur de Hecke T p[END_REF] on the equidistribution of the eigenvalues of Hecke operators T p suggest that the local picture should be fairly clear. This would allow to establish the value distribution results missing in the case of harmonic averages. We plan to address this question in a forthcoming paper. Question 6.9. Can one prove Theorem 5.1 in greater generality for other types of automorphic forms?

The first obvious step would be establishing it for L(f ⊗χ, s). For more general L-functions an appropriate trace formula would be necessary to replace Petersson. The GRH being known in this case, unconditional results should not be very difficult to establish along the lines of this paper, once proper definitions are given. Question 6.11. Establish the properties of M functions in the case of averages with respect to primitive forms. Some peculiarities do arise compared to the case of characters. For example, Ms (z 1 , z 2 ) is no longer holomorphic in s, since the average does depend on s and s. The function is still entire in z 1 , z 2 for fixed s. Establishing its explicit power series expansion, analytic continuation, etc. seems to be of interest. The growth properties of M seem to be much more delicate in our case, since they are proven using local results in the situation of Ihara and Matsumoto. Question 6.12. Write an adelic version of Ihara's and Matsumoto's results, as well as of our results in the setting of modular forms.

This might shed some light on and give a better understanding of the functions M, M , as well as of the relation of the global theory to the local one. One might also hope to be able to deal with the problems related to units in the number field case (c.f. Question 6.1).

Proposition 4. 4 .

 4 (i) There exists a unique positive measure M σ,P of compact support and mass 1 on C ≃ R 2 such that M σ,P (Φ) = T P Φ(g s,P (t P ))d × t P for any continuous function Φ on C. (ii) F M σ,P = Mσ,P (z). (iii) There exists a set of primes P f of positive density such that, for all p ∈ P f , Mσ,p (z) ≪ p,σ (1 + |z|) -1 2 .

Lemma 4 . 5 .

 45 Let ρ > 0 and let F (z) = k≥1 a k z k be absolutely convergent for |z| < ρ + ǫ, ǫ > 0. Let S ⊂ C denote the parametric curve {S(θ)} θ∈[0,1] = {F (re 2πiθ )} θ∈[0,1] . Let D r be the distribution on C = R 2 defined as the direct image of the normalized Haar measure on the circle of radius r in C by F and let Dr = F D r be its Fourier transform. Assume that |a 1 | = 0. Then, if

Corollary 4 . 6 .

 46 Given r > 0, y > 0, there exists C = C(y, r, f ) such that Mσ,Px\Py (z) = O((1 + |z|) -r ) and the function M σ,Px\Py (z) is of class C r for all x ≥ C.

Proposition 4. 8 .

 8 (i) The function Ms (z 1 , z 2 ) is entire in z 1 , z 2 . (ii) We have the Euler product expansion Ms (z 1 , z 2 ) = p Ms,p (z 1 , z 2 ), which converges absolutely and uniformly on Re s ≥ 1 2 + ǫ and |z 1 |, |z 2 | ≤ R, for any ǫ, R > 0. (iii) Mσ (z) = O((1 + |z|) -N ) for all N > 0.

Remark 4 . 10 . 4 . 3 .

 41043 One should be able to write an explicit power series expansion of Ms (z 1 , z 2 ) similar to the one in [11, §4, Theorem M ]. Global results for M σ . Proposition 4.11. The sequence (M σ,Px (z)) x≫0 converges uniformly (as continuous functions) to M σ (z) := F Mσ (-z). Moreover, for a fixed y, the sequence of continuous functions (M σ,Px\Py ) x≫0 converges uniformly to the continuous function M (y) σ = * P\Py M σ,p := F p∈P\Py Mσ,p (-z) , and we have M σ (z) = M σ,Py * M (y) σ .

f

  admits the corresponding partial derivative. The statement now follows from the uniform convergence of M (y) σ,Px\Py to M (y) σ .

σ 2 BM

 2 = * p∈P\Py M σ,p : C M (y) σ (w)e λ|w| 2 dw ≤ K. If D ρ = B(0, ρ), B = B(z, ρ) denote the corresponding closed discs, z / ∈ D ρ , then e λ(|z|-ρ) (y) σ (w)|dw| = B e λ(|z|-ρ) 2 M (y) σ (w)|dw| ≤ B e λ|w| 2 M (y) σ (w)|dw| ≤ K.

  m). Let us first note that the sum doesnot depend on N, since c N z,x (n) = c z,x (n) if (n, N) = 1, and the coefficient c N z,x (n) vanishes, once we have both (x, N) = 1, and (n, N) = 1. This allow us to write M (s) = n,m

r≥1r

  -2σ+2ǫ . Let us now see what happens with the error term. If we put ∆(s) = n,m≥1 n -s m -s x∈J N (n),y∈J N (m) (xy,N )=1 c N z,x (n)c N z ′ ,y (m)∆(x, y), from the Proposition 5.4 together with the estimate τ 3 (n) ≤ τ (n) 3 ≪ ǫ n ǫ , and Lemma 2.1 we conclude that |∆(s)| ≪ ǫ τ (N) 2 log N Nk 5/6 m,n≥1

Lemma 5 . 7 .

 57 Let Re s ≥ 1/2+ǫ. Then for any f

Question 6 . 10 .

 610 What is a function field version of Theorem 5.1?

The two authors were partially supported by ANR Globes ANR-12-JS01-0007-01 and the second author by the Russian Academic Excellence Project '5-100'.

Integral representation.

We introduce the following notation. Let 0 < ǫ ′ < ǫ < 1 2 , s ∈ C with σ = Re s ≥ 1 2 + ǫ, c > max(0, 1 -σ), X ≥ 1 a parameter to be specified later. The symbol ≪ will depend on ǫ, R, and T but this dependence will not be explicitly indicated. As before, we assume only that N is squarefree (and not necessarily prime), and we do not suppose k to be fixed. We will write g to denote g(f, s, z) when no ambiguity is possible.

We use the techniques from [START_REF] Ihara | On log L and L ′ /L for L-functions and the associated "M-functions": connections in optimal cases[END_REF], though it would be possible to employ the approximate functional equations instead, since they are available in our case. First, we establish the analogues of the propositions proven in [11, §2.2].

Lemma 5.6.

(i) For Re s ≥ 1 2 +ǫ we have g = g + -g -, where the holomorphic functions g + and g -are defined by

(ii) The function g + has a Dirichlet series expansion

which is absolutely and uniformly convergent on compacts in C.

Proof. The first statement admits exactly the same proof as the corresponding part of [11, Proposition 2.2.1] with Ihara and Matsumoto's property (A3) being replaced by Lemma 3.2 in our case.

As for the second statement, we have the Dirichlet series expansion

Taking in account that σ + c > 1, we see that

is absolutely and uniformly convergent with respect to Im w on Re w = c. Exchanging the integration and summation and using

we obtain the desired expansion. The absolute and uniform convergence is clear for Lemma 2.

In what follows we will estimate g + on average, which will give the main term, the function g -will on the contrary be estimated individually for each f. The following lemma bounds g -in terms of the parameter X. Question 6.13. What are the arithmetic implications of our results?

The results of Ihara and Matsumoto give us a better understanding of the behaviour of the Euler-Kronecker constants of cyclotomic fields. More generally, since the log case of averaging results for Q concerns, in particular, zeta-functions of cyclotomic fields ζ Q(ζm) (s), which are simply the products of L(s, χ) over primitive Dirichlet characters of conductors dividing m, the results of Ihara and Matsumoto can be seen as a first step in the development of a finer version of the asymptotic theory of global fields from [START_REF] Tsfasman | Infinite global fields and the generalized Brauer-Siegel Theorem[END_REF], that gives non-trivial results for abelian extensions. This is not the case in [START_REF] Tsfasman | Infinite global fields and the generalized Brauer-Siegel Theorem[END_REF], since infinite global fields, containing infinite abelian subfields are asymptotically bad in the terminology of loc. cit.

When one takes averages with respect to primitive forms, the results are close in spirit to the asymptotic study of zeta-functions of modular curves X 0 (N), that can be written as

Establishing a precise relation boils down to answering Question 6.8.

Note that even a cruder version of the asymptotic theory in the spirit of [START_REF] Tsfasman | Infinite global fields and the generalized Brauer-Siegel Theorem[END_REF] has not been developed in this case. In the function field case this was to a significant extent done in [START_REF] Zykin | Asymptotic properties of zeta functions over finite fields[END_REF]. A higher dimensional asymptotic theory in the characteristic zero case is yet to be constructed.


L Marioni
email: luca.marioni@mines-paristech.fr

J Alves

F Bay

E Hachem

EFFECT OF M-EMS ON IN-MOULD TRANSIENT FLOW DURING CONTINUOUS CASTING

Keywords: continuous casting, electromagnetic stirring, mesh adaptation, finite elements

Electromagnetic stirring (EMS) is widely used to increase the efficiency in continuous casting process of steel. In particular, in-mould applications (M-EMS) allow decreasing the free surface fluctuation, controlling the velocity field and decreasing the turbulence of the flow. Since laboratory-scale tests are not fully representative of the process and industrial measurements are both expensive and difficult to carry out, numerical simulation is a strong tool to study and optimize these electromagnetic applications in steel industry. These simulations cannot fully model the process of its multiphysical nature and the simulation of all the phenomena involved would lead to huge computational costs, which is one of the main limits in the current situation. For this reason, this work aims at starting searching a coupling algorithm which could guarantee both computational efficiency and accuracy of the final results.

INTRODUCTION

In continuous casting processes the liquid metal is injected into the mould. The final steel shell is obtained after the solidification which starts in the mould and continues in the strands. Electromagnetic devices such as stirrers and brakers are well known technologies used to improve both the quality of the final product and the casting speed. The main defects of the final shell in terms of micro-structure and surface cracking can be directly related to in-mould phenomena: temperature variation, velocity and pressure of liquid steel, free-surface behaviour and slag entrainment are some of the main causes of defects in the final product. With the purpose of controlling the process and preventing the final product from defects, the process has been enhanced with electromagnetic devices such as stirrers (EMS) and brakers (EMB). The main difference is that the stirrers work under the supply of AC current and produce dynamics magnetic fields. Brakers are permanent magnets or circuits fed by DC current, thus they produce constant magnetic fields. Despite the differences they are based on the same idea that the superposition of a magnetic field to the metal flow will generate Lorentz forces which can drive the flow in accordance with the process' design. The physical phenomena occurring in the mould is a multiphysics problem which includes liquid flow, multiphase analyses, electromagnetic computation, heat transfer and solidification processes where each of these physics depends on the others. Numerical simulations are one of the main tools used to design the process because it is too complex to be solved analytically [START_REF] Fujisaki | 3-D MHD calculation in consideration of free surface, heat transfer and solidification[END_REF]. Also, experimental data are very expensive and difficult to obtain because of the high temperature of the casting process. The main drawback of these simulations is the high computational effort required to simulate the highly multiphysical phenomena including electromagnetic field, multiphase fluid flow, heat transfer and solidification. The study of an efficient coupling is therefore essential to neglect non-relevant aspects of the modelling and obtain fast and reliable simulation algorithms which can be exploited in the industrial world.

SIMULATION TEST CASE

The main purpose of the current work is to create an efficient coupling strategy able to simulate electromagnetic stirring in continuous casting processes. For this reason, a simplified test case has been used. The studied case is a slab casting process and the mould's geometry has been taken from literature (Singh's work [START_REF] Singh | Large eddy simulations of double-ruler electromagnetic field effect on transient flow during continuous casting[END_REF]), but only a quarter of it has been included in the simulation (see figure 1). A simplified electromagnetic stirrer has been positioned away from the mould in the direction of the narrow face and between and under the free surface level. The EMS device is a solenoid fed by a pulsating AC current and produces a horizontal pulsating magnetic field directed from the narrow face in the direction of the nozzle plan. The geometry of the device is simplified since available data on commercial stirrers were not accurate enough to perform a simulation. In the current work we content to simulate a realistic Lorentz force field, since the aim is to study the interaction of the stirrer on the flow more than to obtain specific industrial results. The main data used in the simulation are reported in table I.

MATHEMATICAL MODEL

Governing equations

The potential model has been used to model the electromagnetic field (EMF). Thus, the evolution in time of the EMF is describes by the following system:

{ ( ) ( ) (1)
where and are the free space and material's relative magnetic permeability respectively. is the magnetic vector potential, is the scalar electric potential, is the medium velocity, and is the electrical conductivity. Boundary conditions have been set on the external surface of the inductor by imposing the RMS value of the current density; boundary conditions on the magnetic vector potential have been set as follow:

{ (2)
where is the outward versor normal to the boundary surface. The second equation in system (2) constraints the vector to be tangent to the boundary surface so that all the computed magnetic lines lie totally inside the domain. The convective term can be neglected because the Reynolds magnetic number is , which leads to . For further justification to these assumptions, we refer to [START_REF] Mureau | Magnetohydrodynamics[END_REF]. These conditions ensure that the EMF is not affected by the fluid motion in EMS applications. The term is automatically respected when Nedelec finite elements [START_REF] Nedelec | Mixed finite elements in[END_REF] are used, so the system (1) can be de-coupled; hence the electric potential is computed from the second equation and added into the first one as a source term. The consequent weak formulation of the problem is:

, ( 3
)
where are the shape functions for Nedelec elements. Since the flow is supposed to be over the Curie point and the system reacts linearly with respect to the source term, an harmonic resolution of the time evolution has been adopted. The consequent complex system has been solved via the real-equivalent formulation proposed in [START_REF] Marioni | A new approach to solve complex valued systems arising from the solution of Maxwell equations in the frequency domain through real-equivalent formulations[END_REF] The electromagnetic volumetric force is computed as: [START_REF] Felten | Numerical modelling of electromagnetically-driven turbulent flows using LES methods[END_REF] where the convective term due to fluid motion has been neglected for the aforementioned reasons. Both the time evolving and time averaged Lorentz forces have been considered in two different simulations because the literature does not provide a unique criterion to accept the usage of the averaged field. Such criterion should be dependent on frequency [START_REF] Mureau | Magnetohydrodynamics[END_REF], on the skin depth phenomenon and on the Stuart number [START_REF] Felten | Numerical modelling of electromagnetically-driven turbulent flows using LES methods[END_REF]. The turbulent fluid flow has been modelled by the incompressible Navier-Stokes equation. The Lorentz forces have been considered as a source term. No buoyancy effect was considered, since it is negligible in comparison to inertia and electromagnetic influences. A variational multiscale (VMS) approach [START_REF] Hughes | The variational multiscale method -a paradigm for computational mechanics[END_REF] has been used to solve Navier-Stokes equations. This method allows to implicitly consider the effect of small scale turbulence on the larger eddies developed in the flow. Both velocity and pressure fields have been enriched and stabilized according to [START_REF] Hachem | Stabilized finite element method for incompressible flows with high Reynolds number[END_REF]:

{ ∑ ∑ ∑ (5)
with and the momentum and continuity residuals from the modelling of the fine scale, respectively. and are the density and the viscosity of molten metal, is the pressure and is the volumetric force, whereas and are the shape functions of the weak formulation. System (3) considers the fine scale modelling without explicitly tracking it. The stabilizing terms and (whose theoretical derivation is detailed in [START_REF] Hachem | Stabilized finite element method for incompressible flows with high Reynolds number[END_REF]) enable the model to overcome the instability of the classical formulation arising in convection dominated flows and to deal with the spurious pressure oscillations. No-slip boundary conditions have been imposed at the liquid steel-mould interface and at the free-surface to model the effect of slag friction. Dirichlet boundary conditions on velocity have been imposed at the inlet and outlet surfaces of the mould in respect to the mass conservation. The coupled system has been simulated through the so called two meshes two solvers approach [START_REF] Barna | Numerical Simulations of Mould Electromagnetic Stirring for Round Bloom Strands[END_REF][START_REF] Fujisaki | In-Mold Electromagnetic Stirring in Continuous Casting[END_REF][START_REF] Thomas | State of the Art in Electromagnetic Flow Control in Continuous Casting of Steel Slabs: Modeling and Plant Validation in 6th International Conference on Electromagnetic Processing of Materials EPM[END_REF]. The electromagnetic problem has been solved with Matelec over a domain which included the air and the simplified stirrer. The Lorentz forces have thus been imported in Thercast ® which solves the mechanical problem over the fluid domain. This approach allows us to exploit specific solvers and optimized meshes for each involved physic.

Anisotropic mesh adaptation

In [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF] a posteriori error estimate based on the length distribution tensor approach and the associated edge based error analysis is proposed. This error estimation algorithm is the base for the anisotropic meshing scheme adopted in the current work developed in [START_REF] Coupez | Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF] in the case of Navier-Stokes, multi-field re-meshing. The mesh in the fluid domain is computed as output of an error optimization analysis performed along the edges and referred to user defined fields; we propose the mesh to be adapted according to the following multi-field vector:

( ) { (| |) (| |) } (6)
where is the velocity field, is the distance from the boundary, is the volumetric force and are the nodal indexes. Two different strategies have been used to re-mesh:  Implicit-Static adaptation: the mesh is anistropically adapted to the boundaries and in the weak regions. The obtained mesh is then used as a fixed mesh during the whole computation.  Explicit-Dynamic adaptation: the mesh is dynamically adapted during the computation according to the multicriteria vector (eq. ( 4)) computed from the solution. In figure 2 we can see how the mesh is anisotropically adapted to the flow in the nozzle area. A resume of the meshes used in the simulations is proposed in table II; two overkilled simulations obtained by using different meshing techniques have been performed to check the convergence of results. All the results in the following are taken from the static anisotropic adapted simulation, which is in good agreement with the dynamic re-meshing and the overkilled simulations. The coarse isotropic mesh shows instead sensible discrepancies with the overkilled simulation, because it is not fine enough in the nozzle and in the narrow face thickness.

RESULTS

Flow simulation

The flow developed in the mould is the classical double-roll flow. In figure 3 the magnitude and the streamlines on the average velocity are plotted. The average velocity has been computed in the time interval when a nominally steady state is reached. In figure 3 the abnormal width of the injection flow can be noticed, which is an already observed drawback in symmetric simulations [START_REF] Yuan | Study of computational issues in simulation of transient flow ini continuous casting[END_REF]. This average flow is predictive of the macrophenomena occurring in the mould, but lacks information regarding localized turbulence, which has a strong impact on metallurgical properties. The turbulence leads to short-time subflows detected by the non-average solution; this difference can be measured in term of velocity pulsations, defined as:

√ ∑ ̅ (7)
where is the instantaneous velocity, ̅ is the average velocity, is the number of time-steps and represents the i-th time step. In figure 4 the normalized velocity pulsations during the nominal steady state regime is plotted. We see that a large deviation is present in the main flow region and the maximum deviation occurs where the injected flow impacts on the narrow face; the flow in this region is highly turbulent and this turbulence has effect on the mushy zone. When the stirrer is turned on (figure 4 right) the velocity pulsations is decreased in the meniscus zone, which implies the flow to be closer to the steady state. Despite the low level of the magnetic field in the inner zone of the mould, EMS affects the injection flow as well. The flow is smoother and easily turns in the double-roll pattern, while in the non-stirred simulation the unstable flow close to the narrow face led to dynamic effects in the main injection flow. A second effect of the stirrer is to decrease the global velocity of the flow, which is important especially from a simulation point of view. In figure 5 left, the velocity profile along a vertical line 5 cm far from the central axis is plotted. We see that the horizontal velocity related to the main eddy ( is high, especially when compared to the vertical velocity. This means that the simulated flow and its symmetric one (in the second half of the mould) would have a strong impact in this area, which would create turbulence and lead the global flow to be nonsymmetric. For this reason, the flow with no EMS should be simulated without any symmetry condition. On the contrary, we see that the horizontal velocity in the flow with EMS is lower as well as its ratio with the vertical velocity. This result implies that in the stirred case, the geometrical symmetry plane is more likely to behave like a mechanical symmetry plane. The main effects of the stirrer can be detected close to the narrow face, along the symmetry plane and on the injection flow, but not in nozzle area. In figure 5 right the velocity across the exit central line of the nozzle is plotted and no sensible differences between the stirred and nonstirred solution are predicted. The computational effort of the simulation is not highly affected by the weak coupling between the EM and CFD simulations: in figure 6 an analysis of the computing time is presented. The dynamic mesh adaptation is computationally expensive even if the re-meshing has been performed every 14 time steps. As we can see in figure 7, this choice is not good in terms of optimization. The anisotropic adaptation makes the elements to be stretched in the same direction as velocity: this means that the elementary stiffness contribution is the same for every direction and the final system is well conditioned. In-mould flow is far from being steady, so the velocity field can significantly change at every time step, so the mesh, which was optimized for the solution at a certain time-step, does not fit the new solution field. For this reason the resultant system is bad conditioned, except from the time-step when remashing is done, and a large number of iteration is needed to reach the convergence. On the contrary, the statically anisotropic mesh provides a mesh which is generally good for all the time steps and leads to a better conditioned system: the number of required iterations is decreased by 25% with respect to the isotropic mesh (with the same number of elements).

CONCLUSIONS

In this work a non-industrial EMS application was simulated. The flow has been simulated by using a VMS algorithm in order to take into account the small scale turbulence in an implicit way. The adopted anisotropic mesh adaptation algorithm allows the user to use a low number of elements; it also improves the conditioning of the final system, so fewer iterations are required to solve the problem. A weak coupling scheme is adopted to simulate the coupled electromagnetic-fluid mechanical phenomena. This scheme is computational light, but is able to predict the main features of the simulated application. Finally, a mechanic symmetry condition is considered to be accurate in the stirred flow simulation.

Figure 1 :

 1 Figure 1: Left: Mould geometry (one quarter of the commercial mould); Right: SEN longitudinal section.

Figure 2 :

 2 Figure 2: Isotropic starting mesh (left) and anisotropically adapted mesh (right)

Figure 3 :

 3 Figure 3: Averaged velocity magnitude (left) and streamlines in the double-roll area (right)

Figure 4 :Figure 5 :

 45 Figure 4: Velocity pulsations with no EMS (left) and with EMS (right)

Figure 6 :Figure 7 :

 67 Figure 6: Computational time analysis for dynamic re-meshed simulation (left) and static re-meshed simulation (right)

Table I : Process parameters

 I

	Nozzle port angle
	SEN submergence depth
	Casting speed
	Bulk velocity in SEN cross section
	Thickness of the solidified shell	√	mm
		0.57
	Kinematic viscosity of molten steel
	Density of molten steel
	Resistivity of molten steel
	Resistivity of solids parts
	Inlet Reynolds number
	Inlet magnetic Reynolds number	0.13
	Eddy magnetic Reynolds number	0.15
	Intensity of current
	Frequency
	Simulated mould width
	Simulated mould thickness
	Nozzle port diameter
	Nozzle bore diameter (inner)
	Nozzle bore diameter (outer)

Table II : Meshes details

 II

	Process parametersNumber of elements (including	Details
	boundary elements)
	230258	Static anisotropic mesh generation
	229169	Isotropic non-structured mesh
	230000

(average over the time steps) Dynamically and anisotropically adapted mesh 1136972 Static anisotropic mesh generation (overkilled simulation) 1534290

Isotropic, non-structured, mesh (overkilled simulation)

ACKNOWLEDGEMENTS

The authors give thanks to Transvalor S.A. for funding this research and for the continuous support which allowed the complete exploitation of its software.