Glasnik Matematicki, Vol. 60, No. 1 (2025), 39-58. \( \)

ON THE EULER-STIELTJES CONSTANTS FOR FUNCTIONS FROM THE GENERALIZED SELBERG CLASS

Almasa Odžak and Medina Zubača

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

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

Abstract.   The class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\) is a very broad class of \(L\) functions that contains the Selberg class, the class of all automorphic \(L\) functions and the Rankin–Selberg \(L\) functions, as well as products of suitable shifts of those functions. In this paper, we consider generalized Euler-Stieltjes constants \(\gamma_n(F)\) attached to functions \(F(s)\) from the class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\). These are coefficients in Laurent series expansion of function \(F(s)\) at its pole. We derive an integral representation and an upper bound for these constants. The application of the obtained results in the case of product of suitable shifts of the Riemann zeta function is presented.

2020 Mathematics Subject Classification.   11M26, 11S40

Key words and phrases.   Euler-Stieltjes constants, \(L\)-functions, class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\).

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https://doi.org/10.3336/gm.60.1.03

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