Discrete Measures and the Extended Riemann Hypothesis

Abstract

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined arithmetically on the multiplicative group of positive real numbers to the measure $\zeta_{\mathrm{K}}(2)^{-1}\kappa q dq $, where $\kappa$ denotes the residue of $\zeta_{\mathrm{K}}(s)$ at $s=1$ and $dq$ the Lebesgue measure.