Sami Omar

On the zeros of Dirichlet L-functions

In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet

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

L

The Euler φ-function associated to automorphic L-functions

-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.

The Cardon and Robert Criterion for the Riemann hypothesis

Recently, the Li criterion for the Riemann hypothesis has been extended for a general class of L-functions, so-called the Selberg class S. Omar and K. Mazhouda, Le critere de Li et l’hypothese de Riemann pour la classe de Selberg, J. Number Theory 125(1) (2007) 50–58; Corrigendum et addendum a “Le critere de Li et l’hypothese 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 L-functions and the Hecke L-functions S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet L-functions, LMS J. Comput. Math. 14 (2011) 140–154; On the Li coefficients for the Hecke L-functions, Math. Phys. Anal. Geom. 17(1–2) (2014) 67–81. Basing on the latter numerical experiments, it was conjectured that those coefficients are increasing in n. In this note, we show actually that the Riemann hypothesis holds if and only if the Li coefficients for the Hecke L-functions are increasing in n.

Hashing with elliptic curve L -functions

Abstract In this paper, we define the Euler phi function φπ(n) associated to the automorphic L-function L(s, π), where π is an automorphic irreductible unitary cuspidal representation of GLN(ℚ). Furthermore, we study the asymptotic properties for φπ(n) and prove some results on the summation function ∑n≤xφπ(n).

Le critère de Li et l'hypothèse de Riemann pour la classe de Selberg

Abstract In this paper, we give a reformulation of the Riemann hypothesis for automorphic L-functions in terms of orthogonal polynomials which extends the Cardon and Roberts criterion for the classical case of the Riemann zeta function. Actually, for any principal L-function L(s, π) attached to irreducible cuspidal unitary automorphic representation π of GL(N), we show that the generalized Riemann hypothesis with simple zeros is equivalent to the existence of a certain family of orthogonal polynomials {Pn(z)} such that where ξ(s, π) is the completed automorphic L-function.

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

In this paper, we show that the L-functions attached to Elliptic curves are good candidates to construct a hash function which can be used as a MAC for cryptographic purpose. This is actually due to the fact that they present a one way computation of the coefficients aL. In this work, we present some cryptographic preliminaries and we propose a new protocol for hashing using L-functions of elliptic curves. We also study the security of this protocol and its resistance to collision.