Higher-rank zeta functions and SLn -zeta functions for curves.

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when [Formula: see text] and P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to [Formula: see text]) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

Higher-rank zeta functions for elliptic curves.

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite field [Formula: see text] and any integer [Formula: see text] by[Formula: see text]where the sum is over isomorphism classes of [Formula: see text]-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function of [Formula: see text] if [Formula: see text], is a rational function of [Formula: see text] with denominator [Formula: see text] and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet series[Formula: see text]where the sum is now over isomorphism classes of [Formula: see text]-rational semistable vector bundles V of degree 0 on X, is equal to [Formula: see text] and use this fact to prove the Riemann hypothesis for [Formula: see text] for all n.

Jensen polynomials for the Riemann zeta function and other sequences.

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function [Formula: see text] at its point of symmetry. This hyperbolicity has been proved for degrees [Formula: see text] We obtain an asymptotic formula for the central derivatives [Formula: see text] that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all [Formula: see text] These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.