# An analytic approach to the Riemann hypothesis

## TL;DR

This paper introduces an analytic equation related to the Riemann zeta-function in the critical half-strip, exploring the implications of zeros outside the critical line and their connection to the functional equation.

## Contribution

It presents a new equation for the Riemann zeta-function and analyzes conditions for non-trivial zeros outside the critical line, linking them to the functional equation.

## Key findings

- Equivalence between zeros outside the critical line and solutions to a specific equation.
- Detailed analysis of the solutions' behavior as the variable approaches 1.
- Insights into the functional equation's role in the Riemann hypothesis.

## Abstract

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent to the existence of complex numbers for which equation (5.1) in the paper holds. Such a condition is studied, and the attempt of proving the Riemann hypothesis is found to involve also the functional equation (6.26), where t is a real variable bigger than or equal to 1 and n is any natural number. The limiting behavior of the solutions as t approaches 1 is then studied in detail.