# Discrete Measures and the Extended Riemann Hypothesis

## TL;DR

This paper establishes an equivalence between the Riemann hypothesis for Dedekind zeta-functions and the convergence rate of specific discrete measures related to algebraic number fields, linking number theory and measure theory.

## Contribution

It introduces a novel criterion for the Riemann hypothesis based on the convergence of arithmetically defined discrete measures.

## Key findings

- Riemann hypothesis is equivalent to convergence rate of discrete measures
- Connects number theory with measure convergence in a new way
- Provides a potential new approach to studying the 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.