On \(\tau\)-Li coefficients for Rankin-Selberg \(L\)-functions. (English) Zbl 1383.11065

Bucur, Alina; Ernvall-Hytönen, Anne-Maria; Odžak, Almasa; Roditty-Gershon, Edva; Smajlović, Lejla

Bertin, Marie José (ed.) et al., Women in numbers Europe. Research directions in number theory. Based on the presentations at the WINE workshop, Luminy, France, October 13–18, 2013. Cham: Springer (ISBN 978-3-319-17986-5/hbk; 978-3-319-17987-2/ebook). Association for Women in Mathematics Series 2, 167-190 (2015).

Summary: The generalized \(\tau\)-Li criterion for a certain zeta or \(L\)-function states that non-negativity of \(\tau\)-Li coefficients associated to this function is equivalent to non-vanishing of this function in the region \(\operatorname{Re}s>\tau\). For \(\tau\in[1,2)\) and positive integers \(n\), we define \(\tau\)-Li coefficients \(\lambda_n(\pi\times\pi',\tau)\) associated to Rankin-Selberg \(L\)-functions attached to convolutions of two cuspidal, unitary automorphic representations \(\pi\) and \(\pi'\). We investigate their properties, including the Archimedean and non-Archimedean terms, and the asymptotic behavior of these terms.
For the entire collection see [Zbl 1329.11002].

MSC:

11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

Rankin-Selberg \(L\)-functions; \(\tau\)-Li criterion; unitary automorphic representations