On the representations of $N(T)$ via prime numbers

TL;DR

This paper presents a new representation of the function N(T) using prime number logarithms under the Riemann Hypothesis and compares it with Guinand's existing representation.

Contribution

It introduces a novel prime-based representation of N(T) and analyzes its difference from Guinand's representation under RH.

Findings

New representation of N(T) in terms of prime logarithms

Quantitative comparison with Guinand's representation

Insights into the structure of N(T) under RH

Abstract

In this paper, under the RH, we give a new representation of $N(T)$ in term of sum of the logarithms of the powers of prime numbers and compute the difference between the new representation and Guinand representation.