Two arguments that the nontrivial zeros of the Riemann zeta function ...

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Two arguments that the nontrivial zeros of the Riemann zeta function ...

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Feb 28, 2010 - Abstract. We have used the first 2600 nontrivial zeros Î³l of the Riemann zeta function calculated with 1000 digits accuracy and developed them ...

arXiv:1002.4171v2 [math.NT] 28 Feb 2010

Two arguments that the nontrivial zeros of the Riemann zeta function are irrational Marek Wolf e-mail:[email protected]

Abstract We have used the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin’s constant, what suggests that γl are irrational. Next we have calculated the n-th square roots of the denominators qn of the convergents of the continued fractions obtaining values close to the Khinchin–L`evy constant, again supporting the common believe that γl are irrational.

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Introduction

Bernhard G.F. Riemann has shown [16] that the l.h.s. of the identity valid only for 1: −1 ∞ ∞ Y X 1 1 = 1− s , s = σ + it (1) ζ(s) := s n p p=2 n=1 can be analytically continued to the whole complex plane devoid of s = 1 by means of the following contour integral: Z Γ(−s) (−x)s dx ζ(s) = (2) 2πi ex − 1 x P

where the integration is performed along the path P 6 ' &

Now dozens of integrals and series representing the ζ(s) function are known, for collection of such formulas see for example the entry Riemann Zeta Function in [20] and references cited therein. The ζ(s) function has trivial zeros −2, −4, −6, . . . and infinity of nontrivial complex zeros ρl = βl + iγl in the critical strip: βl ∈ (0, 1). The Riemann Hypothesis (RH) asserts that βl = 21 for all l — i.e. all zero lie on the critical line