Mathematics Department Colloquium: Some numerical observations on Riemann's zeta function

Colloquium | December 12 | 4:10-5 p.m. | 60 Evans Hall

 Yuri Matiyasevich, Russian Academy of Sciences

 Department of Mathematics

During last years the speaker was performing intensive computer calculations searching for new properties of Riemann's zeta function. The talk will present two groups of found phenomena which so far haven't got mathematical explanations.

The first group of phenomena is connected with proposed by the speaker reformulation of the Riemann Hypothesis in terms of special Hankel matrices (their entries are defined via Taylor coefficients of the zeta function). Eigenvalues of these matrices demonstrate interesting patterns, which allowed the speaker to state a conjecture which is stronger than the Riemann Hypothesis.

The second group of phenomena is connected with a particular way to approximate the zeta function by finite Dirichlet series. The plots of the ratios of these approximations surprisingly consist of a number of almost circular arcs each containing a zero of the zeta function.