On Power Sums of Positive Numbers

TL;DR

This paper develops a criterion based on Hausdorff moments and Mergelyan's theorem to determine when a genus 0 entire function has only positive zeros, with applications to the Riemann hypothesis.

Contribution

It introduces a new necessary and sufficient condition for positive zeros of genus 0 entire functions, linking complex analysis and number theory.

Findings

Established a criterion for positive zeros of genus 0 entire functions.

Applied the criterion to the Riemann hypothesis and generalized Riemann hypothesis.

Connected the zeros of entire functions to classical problems in number theory.

Abstract

In this work we establish a necessary and sufficient condition for a genus $0$ entire function $f(z)$ has only positive zeros by applying Hausdorff moment problem and Mergelyan's theorem, the obtained criterion is very much reminiscent of Xian-Jin Li's criterion on the Riemann hypothesis. We also apply this criterion to the Riemann hypothesis and the generalized Riemann hypothesis for certain Dirichlet $L$-series of real primitive characters.