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A note on standard zero-free regions for Rankin–Selberg $L$-functions

Volume 196 / 2020

Satadal Ganguly, Ramdin Mawia Acta Arithmetica 196 (2020), 423-431 MSC: Primary 11M26; Secondary 11F66. DOI: 10.4064/aa190825-1-2 Published online: 31 August 2020

Abstract

Let $ \pi $ be a unitary cuspidal automorphic representation of $\mathrm{GL}_{n}(\mathbb{A}_{F}) $ and $ \tilde {\pi } $ its contragredient. In this short note, we prove a standard zero-free region in the $ t $-aspect for the Rankin–Selberg $ L $-function $ L(s,\pi \times \tilde {\pi }) $, assuming that the fourth powers of the Fourier coefficients $ \lambda _\pi (\mathfrak {p}) $ of $ \pi $ at primes $ \mathfrak {p} $ are bounded on average. For $ n=3 $, we prove the desired fourth moment bound under the assumption of the existence of Langlands $ \operatorname{Sym} ^2 $ lift from $ \mathrm {GL}_3 $ to $ \mathrm {GL}_6 $.

Authors

  • Satadal GangulyTheoretical Statistics and Mathematics Unit
    Indian Statistical Institute
    203 Barrackpore Trunk Road
    Kolkata 700108, India
    e-mail
  • Ramdin MawiaTheoretical Statistics and Mathematics Unit
    Indian Statistical Institute
    203 Barrackpore Trunk Road
    Kolkata 700108, India
    e-mail

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