PROOF OF RIEMANN'S HYPOTHESIS

James Constant

math@coolissues.com

Riemann's hypothesis is proved using Riemann's functional equation

This page is now subject to the author's counterexample at http://www.coolissues.com/mathematics/Riemann/disproof.htm

Introduction

The famous conjecture known as Riemann' s hypothesis1 is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula

x . . . . . . . . . . . . . . . . . . . . . . .x>1 . . . . . . . . . . . . . . . . . (1)

the sum extending to all positive integers n, and the product to all positive primes p. The necessary conditions of convergence hold for complex values of s with real part >1. Considering as a function of the complex variable s, Riemann (1859) proved that satisfies a functional equation

. . . . . . . . . . . . . . . .. . . . (2)

which led Riemann to the theorem that all the zeros of , except those at s=-2,-4,-6, . . . , lie in the strip of the s-plane for which where x is the real part of s. Riemann conjectured that all the zeros in the strip should lie on the line x= ½. Attempts to prove or disprove this conjecture have generated a vast and intricate department of analysis, especially since Hardy (1914) proved that has an infinity of zeros on x= ½ .2 The question is still open in 2008. A prize is available to prove or disprove Riemann's hypothesis.3

 

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