On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L\)-function (Q6591277)

The purpose of this paper is to investigate the generalized Euler-Stieltjes constants attached to the Rankin-Selberg \(L\)-functions associated with two representations. More precisely, 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 \(\mathrm{GL}_m(\mathbb{A}_E)\) and \(\mathrm{GL}_{m'}(\mathbb{A}_E)\), respectively. The generalized Euler-Stieltjes constants of the first kind \(\gamma_ {\pi, \pi'}(k)\) attached to the finite part of the Rankin-Selberg \(L\)-function \(L(s, \pi \times \widetilde{\pi'})\) are defined as coefficients in the Laurent series representation of \(L(s, \pi \times \widetilde{\pi'})\) at \(s = 1 + it_0\):\N\[\NL(s, \pi \times \widetilde{\pi'})=\sum_{k=-\delta(t_0)}^\infty \gamma_ {\pi, \pi'} (k) (s-1-it_0)^k,\N\]\Nwhere \(\delta(t_0)=1\) if and only if \(m=m'\) and \(\pi'\) is isomorphic to \(\pi \otimes |\det|^{it_0}\) for some \(t_0\in \mathbb{R}\); otherwise \(\delta(t_0)=0\). The aim of this paper is to derive a non-trivial upper bound for the coefficients \(\gamma_ {\pi, \pi'} (k)\).