A Brief note on the Riemann hypothesis

TL;DR

This paper explores the Riemann hypothesis by analyzing the Euler's alternating series of the zeta function, proposing a regularized ratio within the critical strip, and providing evidence supporting the hypothesis.

Contribution

It introduces a regularized ratio derived from the Euler's alternating series to investigate the Riemann hypothesis, offering new analytical insights.

Findings

Evidence supporting the Riemann hypothesis

Regularized ratio valid in the critical strip

Analytical approach to the zeta function

Abstract

We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.