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Euler-Stieltjes constants for the Rankin-Selberg $$L$$-function and weighted Selberg orthogonality. (English) Zbl 1397.11080
The logarithmic derivative of the Riemann zeta function $$\zeta(s)$$ has a Laurent series at $$s=1$$ of the form $\frac{\zeta'}{\zeta}(s)=\frac1{s-1}+\sum_{k=0}^\infty\eta_k(s-1)^k.$ The coefficients $$\eta_k$$ can be evaluated $\eta_k=\frac{(-1)^{k-1}}{k!}\lim_{x\to\infty}\left(\sum_{n<x}\frac{\Lambda(n)\log^k(n)}{n}-\frac{\log^{k+1}(x)}{k+1}\right),$ where $$\Lambda(n)$$ is the von Mangoldt function.
In the present paper, a similar formula is derived for the Rankin-Selberg $$L$$-function of two automorphic representations. More precisely, let $$E$$ be a normal number field and $$\mathbb A$$ its ring of adeles. Given two unitary cuspidal representations $$\pi$$, $$\pi'$$ of $$\mathrm{GL}_m({\mathbb A})$$ and $$\mathrm{GL}_{m'}({\mathbb A})$$ respectively, one considers the Rankin-Selberg $$L$$-function $$L(s)=L(s,\pi\times\tilde\pi')$$. Let $$\gamma_{\pi,\pi'}(k)$$ denote the $$k$$-th coefficient of the Laurent series at $$s=1$$. The main result of the paper is an asymptotic formula for the $$\gamma_{\pi,\pi'}(k)$$, strikingly similar to the above, where for instance the von Mangoldt function is replaced by a representation-theoretic counterpart, depending on the representations $$\pi$$ and $$\pi'$$. The result is proven under the condition that at least one of the representations $$\pi,\pi'$$ is self-dual. The results of this paper have been generalised to the case of an arbitrary number field by the authors in [Int. J. Number Theory 13, No. 6, 1363–1379 (2017; Zbl 1429.11094)].

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11S40 Zeta functions and $$L$$-functions
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