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15 January 2002 Upper and lower bounds at s=1 for certain Dirichlet series with Euler product
Giuseppe Molteni
Duke Math. J. 111(1): 133-158 (15 January 2002). DOI: 10.1215/S0012-7094-02-11114-4

Abstract

Estimates of the form $L^{(j)}(s,\mathscr{A})\ll_{\epsilon,j,\mathscr {D_A}}\mathscr {R}^\epsilon_{\mathscr {A}}$ in the range $|s-1|\ll 1/\log \mathscr {R_A}$ for general $L$-functions, where $\mathscr {R_A}$ is a parameter related to the functional equation of $L(s,\mathscr {A})$, can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the $L$-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every $L(s, \pi)$, where $\pi$ is an automorphic cusp form on ${\rm GL}(\mathbf {d},\mathbb {A}_K)$. We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.

Citation

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Giuseppe Molteni. "Upper and lower bounds at s=1 for certain Dirichlet series with Euler product." Duke Math. J. 111 (1) 133 - 158, 15 January 2002. https://doi.org/10.1215/S0012-7094-02-11114-4

Information

Published: 15 January 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1100.11028
MathSciNet: MR1876443
Digital Object Identifier: 10.1215/S0012-7094-02-11114-4

Subjects:
Primary: 11M41
Secondary: 11F66, 11F70

Rights: Copyright © 2002 Duke University Press

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Vol.111 • No. 1 • 15 January 2002
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