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Applications of Generalized Li's criterion equivalent to the Riemann
hypothesis and Generalized Littlewood Theorem about Contour Integrals
involving Logarithm of an Analytical Function to study the Riemann zeta-
function zeroes location

Sergey K. Sekatskii

Laboratoire de Physique de la MatiХre Vivante, IPSB, BSP 408, Ecole
Polytechnique FИdИrale de Lausanne, CH 1015 Lausanne-Dorigny, Switzerland.
E-mail: serguei.sekatski@epfl.ch

We show that Li's criterion equivalent to the Riemann hypothesis [1]
can be generalized in the following way: the sums over Riemann zeta-
function zeroes k_n,a=Sum_rho(1-(1-((rho-a)/(rho+a-1))^n) for any real a
not equal to Ѕ are non-negative if and only if the Riemann hypothesis holds
true [2]. Applying the generalized Littlewood theorem about contour
integrals involving logarithm of an analytical function [3], we show that
this is equivalent to the requirement that all derivatives
1/((m-1)!)*d^m/dz^m((z-a)^(m-1)*ln(\xi(z))) for z=1-a of the Riemann xi-
function for all real a<1/2 are non-negative (correspondingly, the same
derivatives when a>1/2 should be non-positive for this; initial Li's
criterion [1] is a particular case of a=1 or a=0) [2]. The shortcoming
related with the impossibility to use the point z=1/2 for the evaluation of
derivatives is removed analyzing quite recent Voros' criterion [4]: we have
established the polynomials P over (z-1/2)^n such that the derivatives at
z=1/2 of P*ln(\xi(z)) are non-negative if and only if the Riemann
hypothesis holds true; see arXiv: 1407.5758.
In the second part of the talk we present the first application of
this same generalized Li's criterion and prove the following statement: for
any positive integer n there is such real value of an (depending on n) that
for all [pic] and a 1)*ln(\xi(z))) is greater than or equal to 0 does hold true; see arXiv:
1404.4484. Finally, other applications of the generalized Littlewood
theorem to Riemann function studies are given by establishing an infinite
number of equalities involving integrals of the logarithm of the Riemann
zeta-function equivalent to the Riemann hypothesis [3]. In particular, we
show that all earlier known criteria of this kind, viz. Wang, Volchkov,
Balazard-Saias-Yor and Merlini integral equalities, are certain particular
cases of the general approach proposed.

[1] Li X.-E., The positivity of a sequence of numbers and the Riemann
hypothesis, J. Numb. Theor. 65 (1997), 325-333.
[2] Sekatskii S. K., Generalized Bombieri-Lagarias' theorem and generalized
Li's criterion with its arithmetic interpretation, Ukrainian Math. J. 66
(2014), 371-383.
[3] Sekatskii S.K., Beltraminelli S., Merlini D. On equalities involving
integrals of the logarithm of the Riemann [pic]-function and equivalent to
the Riemann hypothesis, Ukrainian Math. J. 64 (2012), 218-228;
arXiv:1006.0323
[4] Voros A., Zeta functions over zeros of Zeta functions and an
exponential-asymptotic view of the Riemann Hypothesis. To appear in RIMS
Kokyuroku Bessatsu; arXiv:1403.4558.