Hardy–Littlewood–Riesz type equivalent criteria for the Generalized Riemann hypothesis

Abstract

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet L-function L(s, χ) is equivalent to the following bound: Let k≥ 1 and ℓ be positive real numbers. For any ϵ> 0 , we have ∑n=1∞χ(n)μ(n)nkexp(-xnℓ)=Oϵ,k,ℓ(x-kℓ+12ℓ+ϵ),asx→∞,where χ is a primitive Dirichlet character modulo q, and μ(n) denotes the Möbius function. This bound generalizes the previous bounds given by Riesz, and Hardy–Littlewood. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.