To Örebro University

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurincikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin & ccaron;ikas in a problem session of a recent workshop on universality.

We study universality properties of the Epstein zeta function E-n(L,s) for lattices L of large dimension n and suitable regions of complex numbers s. Our main result is that, as n -> infinity, E-n(L,s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses a novel combination of an approximation result for Dirichlet polynomials, a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as n -> infinity, E-n(L-1,s) - E-n(L-2,s) is universal in the full half-plane to the right of the critical line as E-n(L,s) varies over all pairs of n-dimensional lattices. Finally, we prove a more classical universality result for E-n(L,s) in the s-variable valid for almost all lattices L of dimension n. As part of the proof we obtain a strong bound of E-n(L,s) on the critical line that is subconvex for n >= 5 and almost all n-dimensional lattices L.

We consider the problem of approximating a function having no zeros on the interior of a set by polynomials having no zeros on the entire set.

We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set K without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets K, where the usual condition that the function is nonvanishing on the boundary can be removed. We conjecture that this version of Mergelyan's theorem might be true for a general set K with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.

We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h >= 2 be an integer. We prove that inf(vertical bar zk vertical bar=1) max(nu=1,...,n)(h) vertical bar Sigma(n)(k=1) Z(k)(nu)vertical bar <= (h - 1 + o(1)root n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order root logn.