Yuri V. Matiyasevich, St. Petersburg Department of the Steklov Institute for Mathematics

Computer experiments for approximating Riemanns zeta function by finite Dirichlet series
In 2011, the speaker began to work with finite Dirichlet series of length N vanishing at N ?1 initial non-trivial zeroes of Riemanns zeta function. Intensive multiprecision calculations revealed several interesting phenomena. First, such series approximate with great accuracy the values of the product (1 ? 2 2^(-s)) zeta(s) ?(s) for a large range of s lying inside the critical strip and to the left of it (even better approximations can be obtained by dealing with ratios of certain finite Dirichlet series). In particular the series vanish also very close to many other non-trivial zeroes of the zeta function (initial non-trivial zeroes know about subsequent non-trivial zeroes). Second, the coefficients of such series encode prime numbers in several ways. So far no theoretical explanation has been given for the observed phenomena. The ongoing research can be followed at http://logic.pdmi.ras.ru/~yumat/personaljournal/finitedirichlet