Selberg Class Research Articles

A study on algebraic differential equations of Gamma function and Dirichlet series

The paper concerns the problem of algebraic differential independence between the gamma function and the function in a certain class F, which contains Dirichlet L-functions, L-functions in the extended Selberg class and some periodic functions. It is proved that the gamma function and the function in F cannot satisfy a class of algebraic differential equations with meromorphic coefficients ϕ having Nevanlinna characteristic satisfying T(r,ϕ)=o(r) as r→∞.

Zeta functions of finite fields and the Selberg class

Zeta functions of finite fields and the Selberg class

TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \sum_{\rho } R(\rho) x^{\rho}, $$ where the sum is over the non-trivial zeros $\rho$ of $\zeta(s)$, $R(x) \in \overline{\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \sum_{\rho } \frac{x^{\rho}}{\rho} $$ has infinitely many zeros in $(1, \infty)$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x < 1$, we have the following non-vanishing theorem: If for an integer $d \ge 1$, $f(\pi \sqrt{d} x)$ has a rational zero in $(0,~1/\pi \sqrt{d})$, then $$ L'(1,\chi_{-d}) \neq 0, $$ where $\chi_{-d}$ is the quadratic character associated to the imaginary quadratic field $K:= \Q(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

English

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve. This allows us to derive a simplified proof of the Zhang and Maynard-Tao result on bounded gaps between primes. We also apply the sieve to other subsequences of the primes and obtain bounded gaps in various settings.

A higher rank Selberg sieve with an additive twist and applications

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve with an ``additive twist'' and provide asymptotic formulas for the same. As an application of this higher rank sieve, we obtain improvements of results of Heath-Brown and Ho-Tsang on almost prime $k$-tuples.

Discrete Universality in the Selberg Class

The Selberg class S consists of functions L(s) that are defined by Dirichlet series and satisfy four axioms (Ramanujan conjecture, analytic continuation, functional equation, and Euler product). It has been known that functions in S that satisfy the mean value condition on primes are universal in the sense of Voronin, i.e., every function in a sufficiently wide class of analytic functions can be approximated by the shifts L(s + iτ ), τ ∈ R. In this paper we show that every function in the same class of analytic functions can be approximated by the discrete shifts L(s + ikh), k = 0, 1,..., where h > 0 is an arbitrary fixed number.

Internal Twists of L-Functions. II

A nonlinear twist F(s; f) of a function F(s) from the extended Selberg class S# is called internal if it belongs to S#. In a previous paper (2014) we showed that, inside a rather general class of nonlinear twists, the internal twists occur only in very special cases; moreover, we gave a first characterization of such twists. Here we complete our previous work by giving a fully detailed description of such internal twists.

Symmetry and Short Interval Mean-Squares

The weighted Selberg integral is a discrete mean-square, that is a generalization of the classical Selberg integral of primes to an arithmetic function $f$, whose values in a short interval are suitably attached to a weight function. We give conditions on $f$ and select a particular class of weights, in order to investigate non-trivial bounds of weighted Selberg integrals of both $f$ and $f\ast\mu$. In particular, we discuss the cases of the symmetry integral and the modified Selberg integral, the latter involving the Cesaro weight. We also prove some side results when $f$ is a divisor function.

A further study on uniqueness in the extended Selberg class of Dirichlet series

The paper concerns the uniqueness problems of $\\mathcal{L}$-functionin the extended Selberg class. Firstly, we show that an$\\mathcal{L}$-function is uniquely determined by the zeros of$\\mathcal{L}-a$ except possibly a small set $G$, where $a\\neq~1$ isa fixed constant. Secondly, we study the uniqueness question of$\\mathcal{L}$-function with a rational movinglinebreak target $R$.

A Weak Converse Theorem for Degree 2 $L$-Functions with Conductor 1

We show that every normalized function of degree 2 and conductor 1 in the extended Selberg class has real coefficients, and certain invariants agree with those of the L -functions of cusp forms for the full modular group. The result may therefore be regarded as a weak converse theorem in such a general setting.

ON ALGEBRAIC DIFFERENTIAL EQUATIONS FOR THE GAMMA FUNCTION AND -FUNCTIONS IN THE EXTENDED SELBERG CLASS

This paper concerns the problem of algebraic differential independence of the gamma function and ${\mathcal{L}}$-functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the ${\mathcal{L}}$-function in a domain $D:=\{z:0&lt;\text{Re}\,z&lt;\unicode[STIX]{x1D70E}_{0}\}$ for a positive constant $\unicode[STIX]{x1D70E}_{0}$.

On relations equivalent to the generalized Riemann hypothesis for the Selberg class

We prove that the generalized Riemann hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $\lambda_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $\Re(\lambda_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $\mathrm{GL}_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-functions.

The joint universality for pairs of zeta functions in the Selberg class

We establish a joint universality theorem for pairs of functions in the Selberg class under certain conditions. This theorem generalizes and unifies several previous results, which were shown individually. We also give further examples of pairs of jointly universal L-functions, and actually extend the known universality theorem for the symmetric power L-function \({L(s, \mathrm{sym}^m f)}\) associated to a holomorphic Hecke eigen cusp form f for \({\mathrm{SL}_{2} (\mathbb{Z})}\) with \({1 \le m \le 4}\).

A note on Li’s coefficients for Hecke L-functions

Recently, the Li criterion for the Riemann hypothesis has been extended for a general class of [Formula: see text]-functions, so-called the Selberg class [S. Omar and K. Mazhouda, Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg, J. Number Theory 125(1) (2007) 50–58; Corrigendum et addendum à “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” [J. Number Theory 125(1) (2007) 50–58], J. Number Theory 130(4) (2010) 1109–1114]. Further numerical computations have been done to verify the positivity of some Li coefficients for the Dirichlet [Formula: see text]-functions and the Hecke [Formula: see text]-functions [S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet [Formula: see text]-functions, LMS J. Comput. Math. 14 (2011) 140–154; On the Li coefficients for the Hecke [Formula: see text]-functions, Math. Phys. Anal. Geom. 17(1–2) (2014) 67–81]. Based on the latter numerical experiments, it was conjectured that those coefficients are increasing in [Formula: see text]. In this note, we show actually that the Riemann hypothesis holds if and only if the Li coefficients for the Hecke [Formula: see text]-functions are increasing in [Formula: see text].

A note on moments of limit log-infinitely divisible stochastic measures of Bacry and Muzy

A multiple integral representation of single and joint moments of the total mass of the limit log-infinitely divisible stochastic measure of Bacry and Muzy [$\textit{Comm. Math. Phys.}$ ${\bf 236}$: 449-475, 2003] is derived. The covariance structure of the total mass of the measure is shown to be logarithmic. A generalization of the Selberg integral corresponding to single moments of the limit measure is proposed and shown to satisfy a recurrence relation. The joint moments of the limit lognormal measure, classical Selberg integral with $\lambda_1=\lambda_2=0,$ and Morris integral are represented in the form of multiple binomial sums. For application, low moments of the limit log-Poisson measure are computed exactly and low joint moments of the limit lognormal measure are considered in detail.

Selberg\u2019s orthonormality conjecture and joint universality of L-functions

In the paper we introduce a new method how to use only an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.

Some remarks on the convergence of the Dirichlet series of L-functions and related questions

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class, indicating a different behavior in the two classes. Next we address two questions about majorants of functions in $\mathcal{S}$, showing links with the distribution of the zeros and with independence results.

Large values of L-functions from the Selberg class

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with Jörn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the “resonance method”, which was introduced by Soundararajan, while the second proof uses methods from Diophantine approximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above.

On Properties of Certain Special Zeros of Functions in the Selberg Class

In this paper, assuming generalized Riemann hypothesis, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2 and location of the first zero with positive imaginary part of function in a certain subclass of the extended Selberg class. We apply our results to automorphic L-functions attached to irreducible unitary automorphic representations of $$GL_N(\mathbb {Q})$$ .

On the mean square of an arithmetical error term of the Selberg class in short intervals

Let [Formula: see text] be a Dirichlet series in the Selberg class of degree [Formula: see text] and let [Formula: see text] be the arithmetical error term of [Formula: see text]. We derive two kinds of the mean square estimates of [Formula: see text] in short intervals of Jutila’s type. Our method is based on the truncated Tong-type formula of [Formula: see text]. We also give several applications of these estimates in the arithmetical problems.