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)].