Effective lower bounds for $L(1,\chi)$ via Eisenstein series

Peter Humphries

Vol. 288 (2017), No. 2, 355–375

DOI: 10.2140/pjm.2017.288.355

Abstract

We give effective lower bounds for $L\left(1,\chi \right)$ via Eisenstein series on ${\Gamma }_0\left(q\right)\setminus ℍ$. The proof uses the Maass–Selberg relation for truncated Eisenstein series and sieve theory in the form of the Brun–Titchmarsh inequality. The method follows closely the work of Sarnak in using Eisenstein series to find effective lower bounds for $\zeta \left(1+it\right)$.

Keywords

Dirichlet $L$-function, lower bounds, Eisenstein series

Mathematical Subject Classification 2010

Primary: 11M20

Secondary: 11M36

Milestones

Received: 14 June 2016

Accepted: 23 December 2016

Published: 28 April 2017