Lower bounds for power moments of L-functions
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Bibliographic record
Abstract
Let π be an irreducible unitary cuspidal representation of GLd(QA). Let L(π, s) be the L-function attached to π. For real k ≥ 0 let Ik(π, T ) = Z T 1 |L (π, 1/2 + it)| dt be the k-th (power) moment of L(π, s). Let απ(p, j) ∈ C (1 ≤ j ≤ d) be the local parameters at prime p. We prove that if |απ(p, j)| ≤ p for unramified primes and for a fixed 0 ≤ γ < 1/4, then Ik(π, T ) T (log T ) 2 for any rational k ≥ 0. As a corollary of this result we establish unconditional lower bounds of the conjectured order of magnitude for the fractional k-th moments of Dirichlet L-functions, modular L-functions, twisted modular L-functions, and Maass L-functions. We derive our results as corollaries of more general theorems related to the lower bounds for fractional moments of analytic functions which have Dirichlet series representations on a complex half-plane. We also establish lower bounds for the k-th moments of Artin L-functions and Dedekind zeta functions of number fields.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.000 |
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| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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