We study the mean square of sums of the kth divisor function d k (n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q → ∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of d k (n) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

For each t ∈ R, define the entire functionwhere Φ is the super-exponentially decaying function(2π 2 n 4 e 9u − 3πn 2 e 5u ) exp(−πn 2 e 4u ).Newman showed that there exists a finite constant Λ (the de Bruijn-Newman constant) such that the zeroes of H t are all real precisely when t ≥ Λ. The Riemann hypothesis is the equivalent to the assertion Λ ≤ 0, and Newman conjectured the complementary bound Λ ≥ 0.In this paper we establish Newman's conjecture. The argument proceeds by assuming for contradiction that Λ < 0, and then analyzing the dynamics of zeroes of H t (building on the work of Csordas, Smith, and Varga) to obtain increasingly strong control on the zeroes of H t in the range Λ < t ≤ 0, until one establishes that the zeroes of H 0 are in local equilibrium, in the sense that locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeroes of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Spohn and Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.

We characterize the limiting distribution of Rudin-Shapiro polynomials, showing that, normalized, their values become uniformly distributed in the disc. This resolves conjectures of Saffari and Montgomery. Our proof proceeds by relating the polynomials' distribution to that of a product of weakly dependent random matrices, which we analyze using the representation theory of SU (2). Our approach leads us to a non-commutative analogue of the classical central limit theorem of Salem and Zygmund, which may be of independent interest.

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