#### Vol. 288, No. 2, 2017

 Recent Issues Vol. 293: 1 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
Effective lower bounds for $L(1,\chi)$ via Eisenstein series

### Peter Humphries

Vol. 288 (2017), No. 2, 355–375
##### 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
Primary: 11M20
Secondary: 11M36