On the Value Distribution of Hurwitz Zeta-Functions at the Nontrivial Zeros of the Riemann Zeta-Function

摘要

We consider the value distribution of Hurwitz zeta-functions $zeta (s,alpha ): = sumnolimits_{n = 0}^infty {frac{1}{{(n + alpha )^s }}} $ at the nontrivial zeros ϱ= β + iγ of the Riemann zeta-function ζ (s):= ζ (s, 1). Using the method of Conrey, Ghosh and Gonek we prove for fixed 0< α< 1 andH ≤T that $$begin{gathered} sumlimits_{T< gamma leqslant T + H} {zeta (varrho ,alpha ) = - left( {Lambda frac{1}{alpha } + sumlimits_{n = 1}^infty {frac{{exp ( - 2pi ialpha n}}{n}} } right)frac{H}{{2pi }}} hfill + O(H exp( - C(log T)^{1/3} ) + T^{1/2 + varepsilon } ) hfill end{gathered} $$ with some absolute constantC > 0 (a similar result was first proved by Fujii [4] under assumption of the Riemann hypothesis). It follows that $frac{{zeta (s,alpha )}}{{zeta (s)}}$ is an entire function if and only if α = 1/2 or α = l. Further, we prove for α ≠ 1/2, 1 the existence of zeros ϱ = β +iγ withT < γ ≤T + T3/4, 1/2 β ≤ 9/10+ ε and ζ(ϱ,α)≠0.