arXiv subjects
Publications and source records attributed to Matthew P Young.
We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown's quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan.
We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.
We show an asymptotic formula for the L^2 norm of the Eisenstein series restricted to a segment of a geodesic connecting infinity and an arbitrary real. For generic geodesics of this form, the asymptotic formula shows that the Eisenstein series satisfies restricted QUE. On the other hand, for rational geodesics, the Eisenstein series does not satisfy restricted quantum ergodicity. As an application, we show that the zero set of the Eisenstein series intersects every such geodesic segment, provided the spectral parameter is large. We also make analogous conjectures for the Maass cusp forms. In particular, we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics.
We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman-Kuznetsov formula for newforms of square-free level and trivial nebentypus. We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin-Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus.
We give a short proof of Levinson's result that more than 1/3 of the zeros of the zeta function are on the critical line.