A Proof of the Riemann Hypothesis

In this study, the Riemann problem is presented with highlights on history of the zeta function.

In this study, the Riemann problem is presented with highlights on history of the zeta function. Thereafter, the real primes, which constitute a novelty, are defined. It allows to generalize the Riemann hypothesis to the reals. A calculus of integral solves the problem. The proof is generalized to the integers by an elementary development.

Mathematics and Statistics, 2021

In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: = 1 2 by assuming that () and (1 −) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis. MSC 2010 Classification: 97I80, 11M41

Mantzakouras Nikos

The Riemann zeta function is one of the most Euler's important and fascinating functions in mathematics. By analyzing the material of Riemann's conjecture, we divide our analysis in the zeta function and in the proof of the conjecture, which has very important consequences on the distribution of prime numbers.

Mantzakouras Nikos

The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.

The American Mathematical Monthly, 2002

2023

Angel Garcés Doz

This paper presents a possible elementary proof of the Riemann hypothesis. We say possible or potential, you have to be very cautious and skeptical of the potential of the evidence presented, is free of a crucial error that invalidate the proof. After several months of extensive review, the author, having found no error we have decided to publish it in the hope that someone will find the error. However, it is considered that the method may be useful in some way. This potential proof uses only the rudiments of analysis and arithmetic inequalities. It includes a first part of the reason why we think that the Riemann hypothesis seems to be true.

This paper is a trial to prove Riemann hypothesis according to the following process. 1. We make one identity regarding x from one equation that gives Riemann zeta function ζ(s) analytic continuation and 2 formulas (1/2 + a ± bi, 1/2 − a ± bi) that show non-trivial zero point of ζ(s). 2. We find that the above identity holds only at a = 0. 3. Therefore non-trivial zero points of ζ(s) must be 1/2 ± bi because a cannot have any value but zero.

In this working paper I try to prove the Riemann hypothesis let the zeta function and the diriklet function ∀ ∈ ℂ ℎ () > 0 () = ∑ (−1) +1 =+∞ =1