Glasnik Matematicki, Vol. 59, No. 1 (2024), 33-49. \( \)

ON THE BOUNDEDNESS OF EULER-STIELTJES CONSTANTS FOR THE RANKIN-SELBERG \(L-\)FUNCTION

Medina Zubača

Department of Mathematical and Computer Sciences, University of Sarajevo, Zmaja od Bosne 35, 71 000 Sarajevo, Bosnia and Herzegovina
e-mail:medina.zubaca@pmf.unsa.ba

Abstract.   Let \(E\) be a Galois extension of \(\mathbb{Q}\) of finite degree and let \(\pi \) and \(\pi'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(\mathbb{A}_E)\) and \(GL_{m'}(\mathbb{A}_E)\), respectively. Let \(\Lambda(s,\pi\times\widetilde{\pi}')\) be a Rankin-Selberg \(L-\)function attached to the product \(\pi\times\widetilde{\pi}'\), where \(\widetilde{\pi}'\) denotes the contragredient representation of \(\pi'\), and let its finite part (excluding Archimedean factors) be \(L(s,\pi\times\widetilde{\pi}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, \pi \times \widetilde{\pi}')\). In this paper, we derive an upper bound for these constants.

2020 Mathematics Subject Classification.   11M26, 11S40

Key words and phrases.   Euler-Stieltjes constants, Rankin-Selberg \(L-\)function

https://doi.org/10.3336/gm.59.1.02

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