RIEMANN'S ANALYTIC EXTENSION DISPROVED

James Constant

math@coolissues.com

Riemann's analytic extension leads to obvious inconsistencies of his zeta function

        Since antiquity, mathematicians have labored to find and apply prime numbers. These numbers extend from 1 to infinity. The Prime Number Theorem (PNT) tells us that the number of primes o_x in the number interval x is approximately x/lnx. The PNT was first conjectured by Gauss and Legendre around 1800 and was proven by Hadamard and independently by Poussin in 1859. The PNT is written simply as o_x~x/lnx and saying that o_x is asymptotic to x/lnx as xtº.¹

        Proof of the PNT was one of the great achievements of analytic number theory. The next step was to reduce or eliminate its approximation. Riemann attacked the problem using a formula devised in 1737 by Euler which states that for every real x>1

                                                                                (1)

where the product runs through all primes. Using the comparison test with exponential series e, Euler's sum and product (1) converge absolutely for all values of x>1. f(x), therefore, has no zeros in the region x>1. This means there are no exact values for primes p_n, in effect confirming the PNT.

        Riemann's Hypothesis (RH) assumes zeros outside the region x>1 and further claims these are connected to primes. By replacing the real variable x in Euler's formula (1) with a complex number s Riemann produced his famous zeta function

                                    s=x+jy                                  (2)                    

which extends from Euler's x>1 domain to the larger domain x>0 and provides zeros in the strip 0<x<1. In 1859, Riemann proved that f(s), satisfies a functional equation which led to the theorem that all the zeroes of, except those at s=-2,-4,-6, . . . , lie in the strip of the s-plane for which 0<x<1 where x is the real part of s. Riemann conjectured that all the zeroes 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 f(s), has an infinity of zeroes on x=½.² The question is still open in 2010. A prize is available to prove or disprove RH.³

        Riemann's analytic extension leads to a number of obvious inconsistencies. First, the equality between the sum and product in Riemann's zeta function (2) is unproven. A comparison of series (1) and (2) leads to the impossibility f(s)=f(x). Since all primes p_n>1 the analytic extension to interval 0<x<1 is irrelevant to determining primes. In interval x>1, the analytic extension would require replacing real prime numbers p_n with imaginary ones p_n+jp_n' in product (2) thus making (2) invalid.

        Second, Riemann's zeta function (2) converges conditionally which allows more than one value for its sum. The sum of a conditionally convergent series can be changed by a suitable rearrangement of the terms of the series. It can be made to converge to any number including zero and can even be made to diverge.⁴ This is not possible for absolutely converging series. Thus, using zeta function (2), one can prove⁵ or disprove RH. A vast number of proofs and disproofs now crowd mathematical analysis.⁶

        Third, the hope of RH was that all the zeros in the strip 0<x<1 should lie on the line x= ½ and that zeros on line x= ½ would predict values of prime numbers p_n. However, the hope is inconsistent with Riemann's zeta function (2). Thus, assuming arguendo that the equality between the sum and product in series (2) is valid, if f(s)=0, one or more of the infinite roots of product (2) must be zero. This happens when p_n^s=p_n^x+jy=p_n^xp_n^jy=0. However, p_n^x can never be 0 and, therefore, x need not necessarily be located on the critical line x= ½. Nor can p_n^jy=cosylnp_n+jsinylnp_n=0 and, therefore, there is no connection between zeros in the strip 0<x<1 and prime numbers p_n.

        Riemann's analytic extension leads to obvious inconsistencies of the zeta function equality of sum and product being invalid; of being impossible to obtain a single sum for the conditionally convergent zeta function; and, even if the zeta function sum and product equality was valid, the impossibility of having a zero root for its product of prime numbers. It is unfortunate that so much effort has kept mathematicians busy trying to prove the irrelevant fact that all the zeros in the strip 0<x<1 should lie on the line x= ½ and so little effort made to recognize the obvious inconsistencies.

² E.T. Bell, The Development of Mathematics, Dover Publications, New York 1972. page 315.

³ Enrice Bombieri's The Riemann Hypothesis (Clay Mathematics Institute) at http://www.claymath.org/prize_problems/riemann.htm

⁴ e.g. see R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York 1947 Vol I pages 372-375

⁵ See Proof of Riemann's Hypothesis at http://www.coolissues.com/mathematics/Riemann/riemann.htm

⁶ See Attempts to Prove Disprove the Riemann Hypothesis at http://en.wikipedia.org/wiki/Riemann_hypothesis#CITEREFWatkins2007

Copyright 2009 by James Constant

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