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Apocalyptic Proof of the Riemann Hypothesis with the Generating Function for Prime Numbers
Version 1
: Received: 19 August 2024 / Approved: 20 August 2024 / Online: 21 August 2024 (04:31:23 CEST)
How to cite: Jalali Koutanaei, M. Apocalyptic Proof of the Riemann Hypothesis with the Generating Function for Prime Numbers. Preprints 2024, 2024081439. https://doi.org/10.20944/preprints202408.1439.v1 Jalali Koutanaei, M. Apocalyptic Proof of the Riemann Hypothesis with the Generating Function for Prime Numbers. Preprints 2024, 2024081439. https://doi.org/10.20944/preprints202408.1439.v1
Abstract
The Riemann zeta function plays a crucial role in number theory and its applications. The Riemann Hypothesis (RH) posits that zeros of other than the trivial ones are located on the line defined by the equation Re(s) =1/2. This paper introduces a novel and straightforward proof of the Riemann Hypothesis. The proof employs a standard method, utilizing the eta function in place of the zeta function, under the assumption that the real part is greater than zero. The equation for the real and imaginary parts of the Riemann zeta function (eta function) is completely separated. Initially, let with s= . The value of the real part is determined by solving the equation, and the process is repeated for identifying the potential roots shared by the two functions, a common value is obtained, leading to a =1/2, which represents the real part of the main root of the function ζ(s). By plotting the zeta function for the obtained a, it is determined that the function has a root only at a =1/2. In another analysis, by identifying the roots of the common potential between the two functions ζ(s) and ζ(1-s), a common value is obtained, which leads to a =1/2. This value represents the real part of the main root of the function ζ(s). Additionally, using a standard method and with the help of two functions ζ(s) and ζ(1-s), the real part of the root of the zeta function is obtained. To create a generator function for prime numbers in terms of b, one can solve the root of the zeta function where it equals one (i.e., and obtain a relationship between b’ and prime numbers. Giving the value of zeta equal to one and s' = a' + ib’, similar to zeta equal to zero, the roots are again placed on the line. Then, by using the zeta function defined by multiplying prime numbers, we arrive at a new meaningful relation between b’ and its corresponding prime number is obtained.
Keywords
Riemann Zeta Function; Number Theory; Riemann's Hypothesis
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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