# Examination of the Proof of Riemann's hypothesis and the conditions for it to hold for Generalized {\zeta}-functions. It holds for Riemann's {\zeta}-function and the resulting distribution of primes

@inproceedings{Mantzakouras2021ExaminationOT, title={Examination of the Proof of Riemann's hypothesis and the conditions for it to hold for Generalized \{\zeta\}-functions. It holds for Riemann's \{\zeta\}-function and the resulting distribution of primes}, author={Nikos Mantzakouras}, year={2021} }

Riemann’s zeta function is one of the most important and fascinating functions of Euler’s mathematics. By analyzing the material of Riemann’s hypothesis, we divide our analysis into the function ζ(z) and the proof of the conjecture, which has very important consequences in terms of the distribution of prime numbers. The proof of Riemann’s hypothesis follows from the simple logic,that when two properties are related, i.e. these equations are zero i.e. ζ(z) = ζ(1− z) = 0 while they have the…

## Figures and Tables from this paper

figure 1 figure 2 table 2 figure 3 figure 4 figure 5 table 5 figure 6 table 6 table 7 figure 7 table 8 figure 8 table 9 figure 9 table 10 figure 10 table 11 figure 11 table 12 figure 12 table 13 figure 13 table 14 figure 14 table 15 figure 15 table 16 table 17 table 18 table 19 table 20 table 21 table 22 table 23 table 24 table 25 table 26

## References

SHOWING 1-10 OF 27 REFERENCES

On Gram's law in the theory of the Riemann zeta function

- Mathematics
- 2009

1. Let {t ν } denote the sequence of the roots of the equation (see [5], p. 261): (1) ϑ(t) = πν, where (see [5], p. 383): (2) ϑ(t) = t 2 ln t 2π − t 2 − π 8 + O 1 t , and ν runs over the set of all…

On the zeros of the Riemann zeta function in the critical strip

- Mathematics
- 1979

We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the…

On the zeros of Riemann's zeta-function on the critical line

- Mathematics
- 2016

Abstract We combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Riemann zeta-function ζ ( s ) are on…

A Geometric Proof of Riemann Hypothesis

- Mathematics
- 2003

Beginning from the formal resolution of Riemann Zeta function, by using the formula of inner product between two infinite-dimensional vectors in the complex space, the author proved the world's…

Solve Polynomial and transcendental Equations with use Generalized Theorem (Method Lagrange)

- Mathematics
- 2021

While all the approximate methods mentioned or others that exist, give some specific solutions to the generalized transcendental equations or even polynomial, cannot resolve them completely.”What we…

On the concavity of Dirichlet's eta function and related functional inequalities

- Mathematics
- 2015

Abstract We prove the strict concavity of Dirichlet's eta function η ( s ) = ∑ j = 1 ∞ ( − 1 ) j − 1 j s on ( 0 , ∞ ) . This extends a result of Wang, who proved in 1998 that η is strictly…

On the zeros of the Riemann Zeta function

- Mathematics
- 2010

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is…

The First 50 Million Prime Numbers

- Mathematics
- 1977

I would like to tell you today about a subject which, although I have not worked in it myself, has always extraordinarily captivated me, and which has fascinated mathematicians from the earliest…

Riemann's Zeta Function

- 2014

Preface This document grew out of lecture notes for a course taught in Cambridge during Lent 2014 by Adam Harper on the theory of the Riemann zeta function. There are likely to be errors, which are…

The Widom-Dyson constant for the gap probability in random matrix theory

- Mathematics, Physics
- 2006

In the bulk scaling limit for the Gaussian Unitary Ensemble in random matrix theory, the probability that there are no eigenvalues in the interval (0,2s) is given by P"s=det(I-K"s), where K"s is the…