uniform distribution theory

Uniform Distribution Theory 8 (2013), no.2, 39–47

uniform distribution theory

DISTRIBUTION OF ZEROS OF L-FUNCTIONS ON THE CRITICAL LINE Kamel Mazhouda — Sami Omar ABSTRACT. In this paper, we prove under the Riemann hypothesis that the zeros of L-functions from the Selberg class are uniformly distributed on the critical line when the conductor is sufficiently large and the degree of these L-functions grows with the analytic conductor.

Communicated by Christian Mauduit

1. Introduction There is an intimate connection between the distribution of the nontrivial zeros of an L-function and the distribution of prime numbers. Many number theorists are interested in the horizontal distribution of zeros. However, there is also much interest in studying the distribution of the imaginary parts of the zeros (the vertical distribution). This phenomenon has been investigated by Lang [2], Tsfasman and Vl˘ adut [7] and Zykin [8] in the case of the zeta functions of number fields and function fields. The extended Selberg class S # consists of Dirichlet series +∞ a(n) , Re(s) > 1 F (s) = ns n=1 satisfying the following hypothesis. • Analytic continuation: there exists a non negative integer m such that (s − 1)m F (s) is an entire function of finite order. We denote by mF the smallest integer m which satisfies this condition. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 11M06,11M26, 11M45, 30D35. K e y w o r d s: L-functions, Selberg class, Weil explicit formula, Riemann hypothesis, Distribution of zeros.

39

K. MAZHOUDA — S. OMAR

• Functional equation: for 1 ≤ j ≤ r, there are positive real numbers QF (the conductor), λj and there are complex numbers μj , ω with Re(μj ) ≥ 0 and |ω| = 1, such that φF (s) = ωφF (1 − s), where φF (s) = F (s)QsF

r

Γ(λj s + μj ).

j=1

The Selberg class S [6] is the set of functions F ∈ S # satisfying the two more following axioms: • Ramanujan hypothesis: a(n) = O(n ); • Euler product: F (s) satisfies +∞ b(pk ) exp F (s) = pks p k=1

with suitable coefficients b(pk ) satisfying b(pk ) = O(pkθ ) for some θ < 12 . It is expected that for every function in the Selberg class the analogue of the Riemann hypothesis holds, i.e, that all non trivial (non-real) zeros lie on the critical line Re(s) = 12 . The degree of F ∈ S is defined by dF = 2

r

λj .

j=1

The logarithmic derivative of F (s) has also the Dirichlet series expression +∞ F − (s) = ΛF (n)n−s , F n=1

Re(s) > 1,

where ΛF (n) = b(n) log n is the generalized von Mangoldt function. Let (2π)dF Q2F , qF = β

where

β=

r

−2λj

λj

,

j=1

be the analytic conductor of F ∈ S. If NF (T ) counts the number of zeros of F (s) ∈ S in the rectangle 0 ≤ Re(s) ≤ 1, 0 < Im(s) ≤ T (according to multiplicities) one can show by standard contour integration the formula NF (T ) = 40

dF T log T + c1 T + O (log T ) , 2π

(1)

DISTRIBUTION OF ZEROS OF L-FUNCTIONS

where

1 log qF − dF log(2π) + 1 2π in analogy to the Riemann-von Mangoldt formula for Riemann’s zeta-function ζ(s), the prototype of an element in S. c1 =

We will call a sequence {Fk (s)}k=1,2,...∞ of L-functions from the Selberg class a family if dk = dFk tends to infinity. A family {Fk (s)}k=1,2,...,∞ of L-functions from the Selberg class is called asymptotically exact if the limit lim

k→+∞

log qFk =η log qFk + log dk

exists. It is called asymptotically bad if η = 0 and asymptotically good otherwise. In this paper, we prove under the Riemann hypothesis that the zeros of L-functions in the Selberg class are uniformly distributed on the critical line Re(s) = 1/2 when the conductor is sufficiently large and the degree of these L-functions grows with the analytic conductor. Our main tool is the Weil explicit formula in conjunction with the theory of distributions. To do so, let {Fk }k=1,2,...,∞ be an asymptotically exact family of functions in the Selberg class of degree dk . For each Fk we define the measure 2π δt(ρ) , (2) HFk = log qFk 1 i

1 2

Fk (ρ)=0

ρ − , and ρ runs over all non trivial zeros of the functions Fn . where t(ρ) = Here δa denotes the atomic (Dirac) measure at a. Assuming the Riemann hypothesis, t(ρ) is real and HFk is a discrete measure on R.

2. The Weil explicit formula The Weil explicit formula will be used in the proof of our main result (Theorem 1) below.

1

[3, 4] Let f : R → C satisfy the following conditions:

• f is normalized, 2f (x) = f (x + 0) + f (x − 0),

x ∈ R. 41


• There is a number b > 0 such that 1 VR f (x)e( 2 +b)|x| < ∞, where VR (.) denotes the total variation on R. • For all 1 ≤ j ≤ r, the function Gj (x) = f (x)e

−ix

Im(μj ) λj

satisfies

Gj (x) + Gj (−x) = 2f (0) + O(|x|). Let F (s) ∈ S. Then,

H(ρ) = mF H(0) + H(1) + 2f (0) log QF

ρ

⎫ ⎧ λ +∞⎪ ⎪ 1− 2j −Re(μj ) λx r ⎨ 2λ G (0) j ⎬ − λx e j j e j dx + Gj (x) + Gj (−x) − − x ⎪ ⎪ x ⎭ ⎩ 1 − e λj j=1 0

∞ ΛF (n) ΛF (n) √ f (log n) + √ f (− log n) , − n n n=1

where

(3)

+∞

f (x)e(s−1/2)x dx et H(ρ) = lim H(s) =

T →∞

ρ

−∞

H(ρ).

|Im(ρ)| 0 is fixed, then |an | ≤ c( )n implies that |b(pv | ≤ c( ) (2v − 1) pv /v. Using the last inequality and that the integral converges uniformly as h(t) ∈ S, we obtain

where

Gj (t) = f (t)e

−it

Im(μj ) λj

H(h) = 2πf (0)η +2

+∞ r h t−

Im(μj ) λj

+ h −t +

Im(μj ) λj

2

lim

ψ

λj 2

+ Re(μj ) + iλj t

log qFk + log dk λj +∞ r ψ + μ + iλ t j j 2 h(t) + h(−t) dt = 2πf (0)η + 2 lim k→+∞ 2 log qFk + log dk j=1−∞ ⎤ ⎡ λ +∞ j r + μ + iλ t ψ j j 2 h(t) + h(−t) ⎣ ⎦ dt. = 2πf (0)η + 2 lim k→+∞ 2 log qFk + log dk j=1 j=1−∞

k→+∞

−∞

Using the Stirling formula, we have ψ(z) = log z + O 44

1 |z|

.

dt

DISTRIBUTION OF ZEROS OF L-FUNCTIONS

Then

⎤ ⎡ λ

ψ 2j + μj + iλj t log d 1 1 k ⎣ ⎦ = lim = (1 − η). lim k→+∞ k→+∞ 2 log qFk + log dk log qFk + log dk 2 j=1 r

Recall that 2πH(0) =

+∞ −∞

ψ(t)dt. Therefore, we deduce

+∞

H = 2πH(0)η + 2πH(0)(1 − η) = 2πH(0) = ψ(t)dt −∞

and Theorem 1 follows.

• Theorem 1 implies that the zeros of an L-functions in the Selberg class are asymptotically uniformly distributed for any asymptotically bad family. • Theorem 1 implies that any fixed interval around Re(s) = 1/2 contains zeros of Fn when qFn is sufficiently large.

• Let F = {Ki }i=1,2,...,∞ be an asymptotically exact family of number fields. For each Ki , we define the measure π δt(ρ) , H= gi ζKi (ρ)=0

where gi = gKi , t(ρ) = (ρ − 1/2)/i, and ρ runs over all non-trivial zeros of the zeta-function ζKi (s). By Theorem 1, we have H = lim Hi = lim HKi i→+∞

i→+∞

exists. The last result was proved also by Tsfasman & Vlˇ adut [7, Theorem 5.2 and §5]. • Let ξN (s) = N

gs/2

(2π)

−gs g

Γ

k−1 k−1 + s LN +s 2 2

where the positive integer weight k > 2, g is the dimension of the space Sk (N, χ), Γ is the Gamma function, the Hecke L-function LN has an Euler product that we omit and χ is a Dirichlet character of modulus N with χ(−1) = (−1)k . Here, Sk (N, χ) is the space of all cusp forms of weight k and character χ for the Hecke congruence subgroup Γ0 (N ) of level N . 45


In particular, f ∈ Sk (N, χ) is holomorphic in the upper half plane and transforms as f [(az + b)/(cz + d)] = χ(d)(cz + d)k f (z)

a b for all 2 × 2 matrices ∈ Γ0 (N ), where c d

a b ∈ SL(2, Z)|c = 0modN . Γ0 (N ) = c d To each newform g(z) we associate the mesure 2π δt(ρ) , Hg = log qg Lg (ρ)=0

where t(ρ) = (ρ − 1/2)/i, and ρ runs over all non-trivial zeros of the zetafunction

k+1 k−1 +3 +3 . Lg (s) and qg = N 2 2 Using Theorem 1 above and assuming the Riemann hypothesis, we deduce that for any family {gi (z)}i=1,2,...,∞ of primitive forms with qgi → +∞, H = lim Hi = lim Hgi i→+∞

i→+∞

exists in the space of measures of slow growth on R and it is equal to the measure with density 1.

The second author is supported by the King Khalid University, research grant No. KKU-S027-33. REFERENCES [1] Barner, K.: On A. Weil’s explicit formula, J. Reine Angew. Math. 323 (1981), 139–152. [2] Lang, S.: On the zeta function of number fields, Invent. Math. 12 (1971), 337–345. [3] Omar, S.—Mazhouda, K.: Le crit` ere de Li et l’hypoth` ese de Riemann pour la classe de Selberg, J. Number Theory 125 (2007), no. 1, 50–58. [4] Omar, S.—Mazhouda, K.: Erratum et addendum a ` l’article, Le crit` ere de Li et l’hypoth` ese de Riemann pour la classe de Selberg [J. Number Theory 125 (2007) 50–58], J. Number Theory 130 (2010), no. 4, 1109–114. [5] Schwartz, L.: Th´ eorie des Distributions, Hermann, Paris, (1966). [6] Selberg, A.: Old and new conjectures and results about a class of Dirichlet series, presented at the Amelfi Conference on Number Theory, September 1989, in: Collected Papers, vol. II, Springer-Verlag, (1991).

46

DISTRIBUTION OF ZEROS OF L-FUNCTIONS ˘ dut, S. G.: Infinite global fields and the generalized BrauerSiegel [7] Tsfasman M. A.—Vla theorem, Moscow Mathematical Journal, 2 (2002), no. 2, 329–402. [8] Zykin, A.: Asymptotic properties of zeta functions over finite fields (preprint). Received November 11, 2012 Accepted January 2, 2013

Kamel Mazhouda Faculty of Sciences of Monastir Department of Mathematics 5000 Monastir Tunisia E-mail : [email protected] Sami Omar Faculty of Sciences of Tunis Department of Mathematics 2092 Campus Universitaire El Manar Tunis, Tunisia E-mail : [email protected]

47