A sequential Riesz-like criterion for the Riemann hypothesis

Abstract

Letck:=∑j−0k(−1)j(kj)(1/ζ(2j+2)). We prove that the Riemann hypothesis is equivalent tock≪k−3/4+εfor allε>0; furthermore, we prove thatck≪k−3/4implies that the zeros ofζ(s)are simple. This is closely related to M. Riesz's criterion which states that the Riemann hypothesis is equivalent to∑k=1∞((−1)k+1xk/(k−1)!ζ(2k))≪x1/4+εasx→+∞, for allε>0.