Τhe Riemann Hypothesis and distribution of the primes

Published: 6 July 2021| Version 4 | DOI: 10.17632/pjcc8z6fpc.4
Contributor:
nikos mantzakouras

Description

THEME "Τhe Riemann Hypothesis and distribution of the primes" Abstract: 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 ζ(z) function and in the proof of the conjecture, which has very important consequences on the distribution of prime numbers. The proof of the Hypothesis of Riemann result from the simple logic, that when two properties are associated, (the resulting equations that based in two Functional equations of Riemann ), if zero these equations, ie ζ(z) = ζ (1-z)= 0 and simultaneously they to have the proved property 1-1 of the Riemann function ζ(z).Thus, there is not margin for to non exist the Re (z) = 1/2 {because ζ (z) = ζ (1-z)=0 and also ζ(z) as and ζ(1-z) are 1-1}.This, as it stands, will gives the direction of all the non-trivial roots to be all in on the critical line, with a value in the real axis equal 1/2.

Files

Institutions

Athens Academy

Categories

Number Theory

Related Links

Article
[1]. A Proof of the Riemann Hypothesis, Murad Ahmed Abu Amr (2016, arXiv.org) [2]. A Geometric Proof of Riemann Hypothesis, Kaida Shi p. 20-45,(2003 arXiv.org) [3].An Exploration of the Riemann Zeta Function and its Applications p 15-60, 2006. [4].Themis Μitsis,(Complex analysis I) p.17-30, http://fourier.math.uoc.gr/~mitsis/notes/undergrad_compl ex.pdf [5].Mantzakouras Nikos Solve Polynomials and Transcendental equations using Mathematica, p.50-80,2012, also in internet…a part https://www.academia.edu/21964173/Solve_Equations_ With_method_Lagrange_ [6].Proof of Hypothesis Riemann,Part I. Mantzakouras Nikos (2017).The zeros of the ζ(z)=0. https://www.academia.edu/26990409/Proof_of_the_Rie
is related to this dataset
Article
[7].Nikolentzoc Poluqronic,R-H p.9-22,2008 [8]A theory for the zeros of Riemann and other 2014, L-functions Guilherme Frand Andre Le Clairy p.38-50 [9]. Titchmarsh E C. The zeros of the Riemann zeta-function. Proc Roy Soc Ser, 1935,151: 234-255 [10].Titchmarsh E C. The zeros of the Riemann zeta-function. Proc Roy Soc Ser, 1936,157: 261-263. [11]. Van de Lune J, te Riele H J J, Winter D T. On the zeros of the Riemann zeta-functionin the critical strip. Math. Com,1986, 46: 667-681. [12].Hardy G H, Littlewood J E. The zeros of Riemann’s zeta-function on the critical line.Math Z, 1921, 10: 283-317. [13]. H.M. Edwards, Riemann's Zeta Function, Academic Press, New York, London, 1974. [14]. D. Zagier, The first 50 million prime numbers", Mathematical Intelligencer (1977), p.7–19. [15].The Mathematical Intelligencer (1977), p 7-19. [16].Andrew Odlyzko,The 21 10 -st zeros of the Riemann zeta function, 1988.& Tables of Zeros. [17] Cloizeaux and Mehta, The Widom–Dyson constant for the gap probability in random matrix theory,2007,p 26-47. [18]. Hugh Montgomery ,The pair correlation of the zeros function,1970. [19].The “data1 & data & NonlinearModelFit Statistic”, have been done with the program mathematica. [20] “The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large heigh”, Xavier Gourdon,version 2004. [21]. On the Gram’s Law in the Theory of Riemann Zeta Function. M.A. Korolev,2010,p4-6.
is related to this dataset

Licence