A note on the zeros of the zeta-function of Riemann

TL;DR

This paper explores the relationship between the sign of Hardy's function and the zero-counting function of the Riemann zeta-function, analyzing definitions and discussing related problems.

Contribution

It explicitly demonstrates how Hardy's function sign depends on the zero-counting function's parity and compares different definitions of this function.

Findings

Hardy's function sign varies with the parity of zero count

Analysis of two definitions of the zero-counting function

Discussion of related open problems

Abstract

It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.