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International Journal of Analysis
Volume 2015, Article ID 980728, 8 pages
http://dx.doi.org/10.1155/2015/980728
Research Article

On Equalities Involving Integrals of the Logarithm of the Riemann -Function with Exponential Weight Which Are Equivalent to the Riemann Hypothesis

1Laboratoire de Physique de la Matière Vivante, IPSB, BSP 408, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
2Research Center for Mathematics and Physics (CERFIM), P.O. Box 1132, 6600 Locarno, Switzerland

Received 30 April 2015; Revised 4 September 2015; Accepted 6 September 2015

Academic Editor: Remi Léandre

Copyright © 2015 Sergey K. Sekatskii et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Integral equalities involving integrals of the logarithm of the Riemann -function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.