Logarithmic moments of characteristic polynomials of random matrices

TL;DR

This paper derives universal explicit formulas for logarithmic moments of characteristic polynomials of random matrices and compares these results to conjectures about the Riemann zeta function, highlighting striking similarities.

Contribution

It extends previous work by providing explicit universal formulas for logarithmic moments in random matrix theory and explores their connection to zeta function conjectures.

Findings

Universal formulas for logarithmic moments derived

Comparison shows striking similarities with zeta function conjectures

Results are independent of specific probability distributions

Abstract

In a recent article we have discussed the connections between averages of powers of Riemann's $\zeta$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $\zeta$-functions, and the correspondence is again striking.