695 research outputs found
Maass waveforms and low-lying zeros
The Katz-Sarnak Density Conjecture states that the behavior of zeros of a
family of -functions near the central point (as the conductors tend to zero)
agrees with the behavior of eigenvalues near 1 of a classical compact group (as
the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo
and Sarnak proved that the behavior of zeros near the central point of
holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal
matrices for suitably restricted test functions . We prove similar
results for families of cuspidal Maass forms, the other natural family of -functions. For suitable weight functions on the space of
Maass forms, the limiting behavior agrees with the expected orthogonal group.
We prove this for \Supp(\widehat{\phi})\subseteq (-3/2, 3/2) when the level
tends to infinity through the square-free numbers; if the level is fixed
the support decreases to being contained in , though we still uniquely
specify the symmetry type by computing the 2-level density.Comment: Version 2.1, 33 pages, expanded introduction on low-lying zeros and
the Katz-Sarnak density conjecture, fixed some typo
Statistics for low-lying zeros of symmetric power L-functions in the level aspect
We study one-level and two-level densities for low lying zeros of symmetric
power L-functions in the level aspect. It allows us to completely determine the
symmetry types of some families of symmetric power L-functions with prescribed
sign of functional equation. We also compute the moments of one-level density
and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.Comment: 45 page
- …