Harmonic analysis of the functions $\tilde{\Delta}(x)$ and $N(T)$

TL;DR

This paper explores the Fourier analysis of the functions (x) and N(T) under the Riemann hypothesis, aiming to deepen understanding of their properties and relationships.

Contribution

It provides a novel Fourier analytical approach to (x) and N(T) assuming the Riemann hypothesis, linking harmonic analysis with prime number theory.

Findings

New Fourier representations of (x) and N(T)

Insights into the oscillatory behavior of these functions

Potential implications for prime distribution studies

Abstract

In this paper, under the Riemann hypothesis, we study the Fourier analysis about the functions $\tilde{\Delta}(x)$ and $N(T)$ .