TY - JOUR

T1 - A universal lower bound for certain quadratic integrals of automorphic L–functions

AU - Clozel, Laurent

AU - Sarnak, Peter

N1 - Publisher Copyright:
© 2024 Elsevier Inc.

PY - 2024/8

Y1 - 2024/8

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jnt.2024.02.018

DO - 10.1016/j.jnt.2024.02.018

M3 - Article

AN - SCOPUS:85189779477

SN - 0022-314X

VL - 261

SP - 252

EP - 298

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -