Regularization for zeta functions with physical applications I

TL;DR

This paper introduces a regularization method for zeta functions, aiming to shed light on the Riemann hypothesis and prime number properties, with future work focusing on zeros and physical applications.

Contribution

It presents a novel regularization technique for zeta functions, providing a new approach to explore the Riemann hypothesis and prime number distribution.

Findings

Regularization makes the Euler product of zeta functions clearer.

Conditions for the Riemann hypothesis are discussed.

Framework for future analysis of zeros and physical applications.

Abstract

We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the Riemann hypotheses by this regularization technique and show conditions to realize them. In part two, we will focus on zeros of the Riemann zeta function and the nature of prime numbers in order to prepare ourselves for physical applications in the third part.