Version 1
: Received: 20 December 2023 / Approved: 21 December 2023 / Online: 21 December 2023 (08:06:19 CET)
How to cite:
Caceres, P. Proof of the Riemann Hypothesis Using the Decomposition Zeta(s) = X(s) – Y(s). Preprints2023, 2023121618. https://doi.org/10.20944/preprints202312.1618.v1
Caceres, P. Proof of the Riemann Hypothesis Using the Decomposition Zeta(s) = X(s) – Y(s). Preprints 2023, 2023121618. https://doi.org/10.20944/preprints202312.1618.v1
Caceres, P. Proof of the Riemann Hypothesis Using the Decomposition Zeta(s) = X(s) – Y(s). Preprints2023, 2023121618. https://doi.org/10.20944/preprints202312.1618.v1
APA Style
Caceres, P. (2023). Proof of the Riemann Hypothesis Using the Decomposition Zeta(s) = X(s) – Y(s). Preprints. https://doi.org/10.20944/preprints202312.1618.v1
Chicago/Turabian Style
Caceres, P. 2023 "Proof of the Riemann Hypothesis Using the Decomposition Zeta(s) = X(s) – Y(s)" Preprints. https://doi.org/10.20944/preprints202312.1618.v1
Abstract
Prime numbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advances in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In that paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function defined by: Where ζ(z) is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of the zeros of RZF, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this: [RH] The real part of every nontrivial zero z* of the RZF is 1/2. Proving the RH is, as of today, one of the most important problems in mathematics. In this paper we will provide a proof of the RH. The proof of the RH will be built following these five parts: PART 1: Description of the RZF PART 2: The C-transformation PART 3: Application of the C-transformation to in Re(z) 0 to obtain ζ(z)=X(z)-Y(z) PART 4: Analysis of the values of z such that X(z)=Y(z), and |X(z)|=|Y(z)|, that equates to ζ(z)=0 Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z) 0 PART 5: We will also prove that all non-trivial zeros of ζ(z) in the critical line of the form are not distributed randomly. There is a relationship between the values of those zeros and the Harmonic function that leads to an algebraic relationship between any two zeros.
Keywords
riemann hypothesis; prime numbers; infinite power series; number theory
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.
