On Power Sums of Positive Numbers

Ruiming Zhang

On Power Sums of Positive Numbers

Classical Analysis and ODEs 2016-02-23 v2

Authors: Ruiming Zhang

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.

Keywords

power sums entire function positive zeros Hausdorff moment problem Mergelyan's theorem Riemann hypothesis generalized Riemann hypothesis

Cite

@article{arxiv.1510.03420,
  title  = {On Power Sums of Positive Numbers},
  author = {Ruiming Zhang},
  journal= {arXiv preprint arXiv:1510.03420},
  year   = {2016}
}