A proof of the Riemann hypothesis using the remainder term of the Dirichlet eta function

TL;DR

This paper claims to prove the Riemann hypothesis by analyzing the remainder term of the Dirichlet eta function and its approximation, establishing a connection between the zeros of eta and the hypothesis.

Contribution

It introduces a novel approach to prove the Riemann hypothesis using the approximation of the eta function's remainder term and its relation to the functional equation.

Findings

Remainder term of eta function can be approximated effectively.

Error between the remainder term and its approximation diminishes as n increases.

The proof links zeros of eta function to the Riemann hypothesis.

Abstract

The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make sure that the error between remainder term and its approximation is reduced as n goes to infinity. According to the Riemann zeta functional equation, if $\eta(\sigma+it)=0$ then $\eta(1-\sigma-it)=0$. In this case, $n$-th partial sum also can be approximated by expression for $n$. Based on this approximation, we prove the Riemann hypothesis.