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Proceedings of the London Mathematical Society Advance Access published online on June 24, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn026
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© 2008 London Mathematical Society

Sampling the Lindelöf Hypothesis with the Cauchy random walk

Mikhail Lifshits

Department of Mathematics and Mechanics
St. Petersburg State University
198504 Stary Peterhof
Russia

Michel Weber

Mathématique (IRMA)
Université Louis-Pasteur et CNRS
7 rue René Descartes
67084 Strasbourg cedex
France
weber@math.u-strasbg.fr

Received 13 April 2007. Revision received 25 September 2007.

We study the behaviour of the Riemann zeta function {zeta}(1/2+it), when t is sampled by the Cauchy random walk. More precisely, let X1, X2, ... denote an infinite sequence of independent Cauchy-distributed random variables. Consider the sequence of partial sums Formula , Formula . We investigate the almost-sure asymptotic behaviour of the system {zeta}(1/2+iSn), n=1, 2, .... We develop a complete second-order theory for this system and show, by using a classical approximation formula of {zeta}(·), that it behaves almost like a system of non-correlated variables. Exploiting this fact in relation with the known criteria for almost-sure convergence allows us to prove the following almost-sure asymptotic behaviour: for any real b>2 it is true that Formula .

2000 Mathematics Subject Classification 11M06, 60G50 (primary), 60F15 (secondary).

The work of the first author was supported by grants NSh.422.206.1, RFBR 05-01-00911, and INTAS 03-51-5018.


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