Duke Mathematical Journal

Upper and lower bounds at s=1 for certain Dirichlet series with Euler product

Giuseppe Molteni

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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.

Article information

Duke Math. J., Volume 111, Number 1 (2002), 133-158.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F70: Representation-theoretic methods; automorphic representations over local and global fields


Molteni, Giuseppe. Upper and lower bounds at s =1 for certain Dirichlet series with Euler product. Duke Math. J. 111 (2002), no. 1, 133--158. doi:10.1215/S0012-7094-02-11114-4. https://projecteuclid.org/euclid.dmj/1087575009

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