Fundamental properties of symmetric square \(L\)-functions. I.
The Rankin-Selberg zeta-function \(Z(s)\), \(s=\sigma+it\), of weight \(k\) with normaled coefficients \(| a_{n^2}| \leq (d(n))^2n^{k-1}\), is defined by \[ Z(s)=\zeta(2s)\sum_{n=1}^{\infty}a_{n^2} n^{1-k-s}=\sum_{n=1}^{\infty}c_n'n^{-s}, \] the series being absolutely convergent in the half-plane \(\sigma>1\), and can be continued as a meromorphic function to the whole complex plane with a simple pole at \(s=1\). The author continues the works of \textit{A. Ivić} [Acta Arith. 56, No. 2, 135--159 (1990; Zbl 0659.10053)], \textit{K. Matsumoto} [Number theory. Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, May 31--June 4, 1999. Berlin: de Gruyter. 201--221 (2001; Zbl 0972.11075)] and other mathematicians in the direction of the investigation of the upper bound for the mean-square of the function \(Z(s)\) in the critical strip. For example, unconditionally it is proved that, for every \(\epsilon>0\), \( \int_{0}^{T}| Z(1/2+it)| ^2 dt \ll T^{11/6+\epsilon} \), and that, under the Lindelöf hypothesis for the Riemann zeta-function, \( \int_{0}^{T}| Z(1/2+it)| ^2 dt \ll T^{3/2+\epsilon} \). If, for some \(\kappa >0\) and all \(t \geq 10\), the estimate \[ \zeta(1/2+it)\ll t^\kappa(\log t) \] is true, then the following results are proved: 1. For \(1/2 \leq \sigma\leq 3/4\), \[ \int_{T}^{2T}| Z(\sigma+it)| ^2 dt \ll T^{1/2+(4\kappa+2)(1-\sigma)}(\log T)^{17(3-4\sigma)+2}; \] 2. For \(3/4<\sigma \leq 1\), \[ \int_{T}^{2T}| Z(\sigma+it)| ^2 dt =T\sum_{n=1}^{\infty}{c_n'}^2n^{-2\sigma}+O(T^{\theta(\sigma)+\epsilon}), \] where \(\theta(\sigma)=5/2-2\sigma\) if \(3/4<\sigma\leq \vartheta\), and \(\theta(\sigma)=(30(1-\sigma))/(17-5 \nu-10\sigma)\) if \(\vartheta \leq \sigma<1\), \(\nu=5/(2(6-4\kappa))\), and \(\vartheta=(29-10 \nu+\sqrt{100 \nu^2+420 \nu-159}))/40\).
scientific article; zbMATH DE number 1839930
scientific article