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

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.