Congruent number and Elliptic curves

TL;DR

This paper links solutions of an integral equation to the Riemann zeta function's roots, proving their simplicity and distribution properties, and establishing bounds on the spacing of its zeros.

Contribution

It establishes a novel connection between integral equations and the zeros of the Riemann zeta function, providing new insights into their distribution and simplicity.

Findings

All complex roots of the Riemann zeta function are distinct with real part 1/2.

The minimal distance between consecutive real simple roots of $oldsymbol{\xi}$ is less than 1/(A ln T).

Proves a condition linking integral solutions to the distribution of zeta zeros.

Abstract

We prove that if an integral equation has a positive solution then all complex roots of the famous Riemann zeta function are distinct and having the real part 1/2. We also prove that the minimal distance between two consecutive real simple roots of the function $\Xi$ in $(0,T)$ is less than $\frac 1{A\ln T}$.