Closed-form formula of Riemann zeta function and eta function for all non-zero given complex numbers via sums of powers of complex functions to disprove Riemann Hypothesis

TL;DR

This paper presents a closed-form formula for the Riemann zeta and eta functions at non-zero complex numbers, claiming to disprove the Riemann Hypothesis by showing the functions are divergent and have infinitely many zeros.

Contribution

It introduces an explicit identity for sums of powers of complex functions, providing a new perspective on the divergence and zeros of the Riemann zeta and eta functions.

Findings

Riemann zeta function is entirely divergent at non-zero complex numbers.

The formula indicates multiple solutions for the zeta function, challenging its traditional properties.

Zeros of the zeta function include some that are also zeros of the Riemann Hypothesis.

Abstract

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no unique solution for any given non-zero complex numbers which means Riemann zeta function is entirely divergent. Infinitely many zeros of Riemann zeta function produced unfortunately those zeros also gives us non-zero values of Riemann zeta function. Among those zeros some of them are zeros of Riemann hypothesis. The present paper also discussed on eta function(alternating Riemann zeta function) with exactly the same behavior as Riemann zeta function.