Matti Jutila | Searcharxiv

arXiv2005

Uniform Bound for Hecke L-Functions

Our principal aim in the present article is to establish a uniform hybrid bound for individual values on the critical line of Hecke $L$-functions associated with cusp forms over the full modular group. This is rendered in the statement that for $t\ge0$ $$ \eqalignno{H_j(\txt{1\over2}+it)&\ll(κ_j+t)^{1/3+ε},& (1.1)\cr H_{j,k}(\txt{1\over2}+it)&\ll (k+t)^{1/3+ε},&(1.2)\cr} $$ with the common notation to be made precise in the course of discussion. Talks on this and relevant results were delivered by the authors at MF Oberwolfach on September 20, 2004, and at General Assembry of Math. Soc. Japan on March 30, 2005.

math.NT↗

arXiv2003

On the moments of Hecke series at central points II

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{κ_j\le T}α_j H_j(1/2) = ({T\overπ})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$ and the Bruggeman--Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^ε)$.

math.NT↗