Reformulation of the Li criterion for the Selberg class

2014/05/28 by Mazhouda, Kamel
#11M06 #11M26 #11M36 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1405.7354

Abstract

Let F be a function in the Selberg class \mathcal S and a be a real number not equal to 1/2. Consider the sum λ_F(n,a)=∑_ρ[1-((ρ-a)/(ρ+a-1))ⁿ], where ρ runs over the non-trivial zeros of F. In this paper, we prove that the Riemann hypothesis is equivalent to the positivity of the "modified Li coefficient" λ_F(n,a), for n=1,2,.. and a<1/2. Furthermore, we give an explicit arithmetic and asymptotic formula of these coefficients.

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