Path integrals as discrete sums

Will a physicist prove the Riemann hypothesis?

In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann hypothesis. In the second part we review some physical problems related to this hypothesis: the Polya-Hilbert conjecture, the links with random matrix theory, relation with the Lee-Yang theorem on the zeros of the partition function and phase transitions, random walks, billiards etc.

Studying open quantum systems by means of a deterministic approach to approximate functional integration

Representation of propagator for open quantum systems in the form of a double functional integral with respect to conditional Wiener measure is proposed. It allows one to apply the approximate formulas exact for functional polynomials of a certain power to calculation of such integrals. Within this deterministic approach the problem is reduced to evaluation of usual (Riemann) integrals of low multiplicity. The formulas are in fact the basis of a numerical method of studying time evolution of the systems. The features of the method are discussed and some examples of calculations are given.