A note on the Selberg class (degrees 0 and 1)

We give a classification of the Dirichlet polynomials in the Selberg class as entire quotients of Dirichlet series with periodic coefficients and the Riemann zeta-function. Received: 2 November 2001     

The Li criterion and the Riemann hypothesis for the Selberg class II

In this paper, we prove an explicit asymptotic formula for the arithmetic formula of the Li coefficients established in Omar and Mazhouda (2007) [10] and Omar and Mazhouda (2010)[11]. Actually, for any function F(s) in the Selberg class S, we have

     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

On a question of Balazard and Saias related to the Riemann hypothesis

This note shows that a question of Balazard and Saias, concerning the coefficients of certain orthogonal projections in Hilbert space, is equivalent to the Riemann hypothesis.     

The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.     

On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψk(x;q,a) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression . This theorem not only gives the asymptotic formula for ψk(x;q,a) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.     

Computation of Weng's rank 2 zeta function over an algebraic number field

In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions.     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

On the Shintani zeta function for the space of binary quadratic forms

Partially supported by NSF Grant DMS-8803085, DMS-8610730     

A criterion for the Generalized Riemann hypothesis

In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.     

General linear independence of a class of multiplicative functions

The notion of global non-equivalence of a set of multiplicative functions is introduced. The linear independence of a set of globally inequivalent multiplicative functions with respect to the ring where r is a slowly varying function is proved. Applications to families of Artin L-functions are given. Received: 4 July 2003     

On some expansions for the Euler Gamma function and the Riemann Zeta function

In the present paper we introduce some expansions which use the falling factorials for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.