International Journal of Analysis

Volume 2015, Article ID 980728, 8 pages

http://dx.doi.org/10.1155/2015/980728

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

^{1}Laboratoire de Physique de la Matière Vivante, IPSB, BSP 408, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland^{2}Research 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.

#### 1. Introduction

In recent papers [1, 2] we analyzed certain contour integrals involving the logarithm of the Riemann zeta-function and have established an infinite number of equalities of the type which were proven to be equivalent to the Riemann hypothesis (RH; is the Riemann zeta-function; see, e.g., [3] for definitions and discussion of the general properties of this function). In particular, it was shown that all earlier known equalities of this type, that is, those of Wang [4], Volchkov [5], Balazard et al. [6], and one of us [7], are certain particular cases of our general approach elaborated in [1].

In this paper we establish new integral equalities equivalent to RH. We use exponential weight functions, and, in our opinion, the resulting equations are especially interesting. In particular, we were able to rigorously estimate the possible contribution of the Riemann function zeroes nonlying on the critical line which were shown to be extremely small, for example, smalle8r than nine milliards of decimals for the “maximal possible”; in a sense (see below), weight function .

#### 2. Integral Equalities with Exponential Weight Function Equivalent to the Riemann Hypothesis

The main tool for our work here is the following generalized Littlewood theorem about contour integrals involving logarithm of an analytical function.

Theorem 1 (the generalized Littlewood theorem). *Let denote the rectangle bounded by the lines where and let be analytic and nonzero on and meromorphic inside it; let also be analytic on and meromorphic inside it. Let be the logarithm defined as follows: one starts with a particular determination on and obtains the value at other points by continuous variation along = const from . If, however, this path would cross a zero or pole of , one takes to be accordingly as one approaches the path from above or below. Let also be the logarithm defined by continuous variation along any smooth curve fully lying inside the contour which avoids all poles and zeroes of and starts from the same particular determination on . Suppose also that the poles and zeroes of the functions , do not coincide.**Then , where the sum is over all which are poles of the function lying inside , all which are zeroes of the function counted taking into account their multiplicities (i.e., the corresponding term is multiplied by for a zero of the order ) and which lie inside , and all which are poles of the function counted taking into account their multiplicities and which lie inside . The assumption is that all relevant integrals in the right-hand side of the equality exist.*

*Proof. *Our proof closely follows the well-known proof of the Littlewood theorem (or lemma) given, for example, in [8, p. 133]. Consider first the function where is a point of the rectangle. Let be the contour obtained by describing in the positive direction from (, ) as far as (), then the straight line as far as , then a circle of radius about , and then returning along and the rest of to the starting point; see Figure 1. The only poles of in are those of the function , so that . The integral round the small circle tends to zero with the radius; thus we have , where and are the values of on the two paths joining to . Hence we obtain from by passing in the negative direction round a simple zero of at ; we have and, correspondingly, , where we introduce a notation to distinguish this function from . The general case now easily follows by addition of terms corresponding to the various poles and zeroes of .