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

Abstract

Problem statement: In the prime number the Riemann zeta function is unquestionable and undisputable one of the most important questions in mathematics whose many researchers are still trying to find answer to some unsolved problems such as Riemann Hypothesis. In this study we proposed a new method that proves the analytic extension 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 consequently provides the required analytic extension of the zeta function to the real part greater than zero 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 its 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.