Proof of Analytic Extension Theorem for Zeta Function Using Abel Transformation and Euler Product

Zacharie Zacharie

doi:10.3844/jmssp.2010.294.299

Abstract

Problem statement: In the prime number the Riemann zeta function is u nquestionable and undisputable one of the most important questions in mathematics whose many researchers are still trying to find answer to some unsolved problems suc h as Riemann Hypothesis. In this study we proposed a new method that proves the analytic exte nsion theorem for zeta function. Approach: Abel transformation was used to prove that the extension theorem is true for the real part of the complex variable that is strictly greater than one and cons equently provides the required analytic extension o f the zeta function to the real part greater than zer o and Euler product was used to prove the real part of the complex that are less than zero and greater or equal to one. Results: From this proposed study we noted that the real values of the complex variable are lying between zero and one which may help to understand the relation between zeta function and i ts properties and consequently can pay the way to solve some complex arithmetic problems including the Riemann Hypothesis. Conclusion: The combination of Abel transformation and Euler product is a powerful tool for proving theorems and functions related to Zeta function including other subjects such as radio atmospheric occultation.

Highlights

There are a number of mathematical functions called “Zeta function” named for their customary symbol, the Greek letter ζ
The most famous is the Riemann Zeta Function, for its involvement in the Riemann Hypothesis, which is highly important in Prime Number Theory (PNT)
The Riemann Zeta function ζ(s) is the function of a complex variable s initially defined by the following infinite series:

Summary

INTRODUCTION

There are a number of mathematical functions called “Zeta function” named for their customary symbol, the Greek letter ζ. Since the 1859 study of Bernhard Riemann, (Castellanos, 1988; David, 1998), it has become standard to extend the definition of ζ(s) to a complex values s; by showing that the series is converges for all complex s whose real part Re(z) is greater than one and defines an analytic function of the. Where: p1, ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅,pr = Distinct prime n1, ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅,nr = Natural numbers and using the fundamental theorem of arithmetic, that is to say every integer can be written in essentially only one way as a product of primes to conclude that the sum is : Euler used this formula principally as a formal identity and principally for integer values of s.

MATERIALS AND METHODS

DISCUSSION

CONCLUSION