On common zeros of L-functions

We will prove that two L-functions L ₁, L ₂ satisfying the same functional equation must be identically equal if sufficiently many zeros of L ₁ are also zeros of L ₂.     

Inequalities for the zeros of dirichlet L-functions

Let χ denote a primitive, Dirichlet character to the modulus q>i and let L(s,χ) be the corresponding Dirichlet L-series defined by L(s,χ) = ∑χ(n)n^?s,s = σ+it, for σ>0. It is of interest to know where the zeros of L(s,χ) are located, since the location of these zeros would yield important results in number theory. In this paper, we show that the spectrum of each member of a certain class of Hermitian matrices leads to an explicit zero-free region for L(s,χ).     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

Low-lying zeros of number field L-functions

TextOne of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec (2003) [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithmetic of the fields. We then derive the first lower order term of the 1-level density, and see the arithmetic enter.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=zpb-gu3G8i0.     

Joint universality for Dirichlet L-functions and zeros of their linear combinations in the $$\chi $$-aspect

We establish a joint universality theorem for Dirichlet L-functions in the character aspect. This is an extension of a result obtained by Bagchi and Gonek independently, and is an analogue of the joint universality for Dirichlet L-functions in the t-aspect. Zeros of linear combinations of Dirichlet L-functions in the t-aspect have been investigated by various authors. Using our joint universality theorem, we investigate zeros of such combinations from a new viewpoint. More precisely, we show that for any region \(\Omega \) in the strip \(1/2< \mathrm {Re}\,s <1\), any non-zero meromorphic functions \(H_1 (s), \dots , H_r(s)\) on \(\Omega \) with \(r \ge 2\) and any positive integer N, there exist a positive integer m and Dirichlet characters \(\varphi _1, \dots , \varphi _r \bmod m\) such that \(\sum _{j=1}^r H_j (s) L(s, \varphi _r)\) has at least N distinct zeros in \(\Omega \).     

On the mean values of Dirichlet L-functions

We study the 2kth power moment of Dirichlet L-functions L(s,) at the centre of the critical strip , where the average is over all primitive characters (mod q). We extend to this case the hybrid Euler–Hadamardproduct results of Gonek, Hughes and Keating for the Riemannzeta function. This allows us to recover conjectures for themoments based on random matrix models, incorporating the arithmeticalterms in a natural way.     

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factoriz...     

On the Li coefficients for the Dirichlet L-functions

In this paper, we prove under the Riemann hypothesis that the Li coefficients for the Dirichlet $L$ -functions $\lambda _{\chi }(n)$ are increasing in $n$ . We also prove unconditionally that the first Li coefficients are increasing using the Bell polynomials. Furthermore, we give a probabilistic interpretation and describe another method differently as stated in Omar et al. (LMS J Comput Math 14:140–154, 2011) to compute them.     

Real zeros of Eisenstein series and Rankin-Selberg L-functions

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $\tfrac{1}{5}$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.     

On discrete mean values of Dirichlet L-functions

Czechoslovak Mathematical Journal - Let χ be a nonprincipal Dirichlet character modulo a prime number p ? 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left(...     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

The third moment of quadratic Dirichlet L-functions

We study the third moment of quadratic Dirichlet $L$ -functions, obtaining an error term of size $O(X^{3/4 + \varepsilon })$ .