A universal lower bound for certain quadratic integrals of automorphic L–functions

Laurent Clozel, Peter Sarnak

Research output: Contribution to journalArticlepeer-review

Abstract

Let π be a cuspidal unitary representation od GL(m,A) where A denotes the ring of adèles of Q. Let L(s,π) be its L-function. We introduce a universal lower bound for the integral ∫−∞+∞|[Formula presented|2dt where s is equal to 0 or is a zero of L(s) on the critical line. In the main text, the proof is given for m≤2 and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series L(s,π). In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.

Original languageEnglish (US)
Pages (from-to)252-298
Number of pages47
JournalJournal of Number Theory
Volume261
DOIs
StatePublished - Aug 2024
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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