Moments of products of automorphic \(L\)-functions
type: Item
instance of: scholarly article
title: Moments of products of automorphic \(L\)-functions
zbMATH Open document ID: 1305.11037
author: Micah B. Milinovich
author: Caroline L. Turnage-Butterbaugh
published in: Journal of Number Theory
publication date: 2014-06-25T00:00:00Z
review text: Understanding the behavior of the Riemann zeta function, and more generally, zeta and \(L\)-functions possessing an Euler product on the critical line \(\mathrm{Re}(s)=1/2\) is one of fundamental problems of analytic number theory. For primitive \(L\)-functions belonging to the Selberg class and for automorphic \(L\)-functions attached to irreducible cuspidal automorphic representations of \(\mathrm{GL}_m\) over \(\mathbb{A}\) with unitary central character it is conjectured that for any \(k>0\) there exist a constant \(C(k,L)\) such that \[ \int_{0}^{T}\left| L(1/2 +it)\right|^{2k} \sim C(k,L) T (\log T)^{k^2}, \] as \(T \to \infty\). For degree-one functions in the Selberg class, i.e., the Riemann zeta function and Dirichlet \(L\)-functions the above conjecture is known to hold for \(k\in\{1,2\}\). For degree-two functions, the conjecture is known to hold with \(k=1\) for certain \(L\)-functions. Actually, it is expected that the values of distinct primitive \(L\)-functions on the critical line are uncorrelated. Therefore, it is expected that for any \(r\) distinct primitive \(L\)-functions \(L_1, \dots, L_r\) and any \(k_1, \dots, k_r >0\) there exist a constant \(C=C(k_1,\dots,k_r, L_1, \dots, L_r)\) such that \[ \int_{0}^{T} \left|L_1(1/2 + it) \right|^{2k_1} \times \cdots \times \left|L_r(1/2 + it) \right|^{2k_r} dt \sim C T (\log T)^{k_1^2 + \dots + k_r^2} \] In the paper under review, the authors prove, under the generalized Riemann hypothesis, upper bounds for moments of products automorphic \(L\)-functions, in support of the product conjecture. Namely, they prove that for \(L\)-functions \(L(s,\pi_1), \dots, L(s,\pi_r)\) attached to distinct irreducible, unitary cuspidal automorphic representations \(\pi_1, \dots, \pi_r\) of \(\mathrm{GL}_{m_1}(\mathbb{Q}), \dots, \mathrm{GL}_{m_r}(\mathbb{Q})\) respectively, one has \[ \int_{0}^{T} \left|L(1/2 + it, \pi_1) \right|^{2k_1} \times \cdots \times \left|L(1/2 + it, \pi_r) \right|^{2k_r} dt \ll T (\log T)^{k_1^2…
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 6308467
zbMATH Keywords: automorphic \(L\)-function
zbMATH Keywords: Dedekind zeta function
zbMATH Keywords: mean values
zbMATH Keywords: moments
reviewed by: Lejla Smajlovic
MaRDI profile type: Publication
full work available at URL: j.jnt.2013.12.012
OpenAlex ID: W1995160613
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DOI: 10.1016/J.JNT.2013.12.012
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description: scientific article; zbMATH DE number 6308467