**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 hypothesis^{1} is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula

. . . . . . . . . . . . . . . . . . . . . . .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}** **

**Update at https://www.smashwords.com/books/view/519332**

**By same author: ** **http://www.coolissues.com/mathematics/sameauthor.htm**