The mean values and the universality of Rankin-Selberg \(L\)-functions (Q2710126)

The author studies the analytic behaviour of the Rankin-Selberg \(L\)-function NEWLINE\[NEWLINE Z(s)=\zeta(2s)\sum_{n=1}^{\infty}a(n)^2 n^{1-\kappa-s} =\sum_{n=1}^{\infty}c_n n^{-s}\qquad(\Re s > 1), NEWLINE\]NEWLINE where \(a(n)\) is the \(n\)th Fourier coefficient of a holomorphic normalized Hecke-eigen cusp form of weight \(\kappa\) with respect to the full modular group, and \(\zeta(s)\) is the Riemann zeta-function. Theorem 1 in this paper shows that the analogy with Voronin's universality theorem for \(\zeta(s)\) holds for \(Z(s)\) in the strip \(3/4< \sigma = \Re s < 1\). This result is obtained by applying the techniques from probabilistic number theory. Theorem 2 gives several mean square results on \(Z(s)\). Standard reflection technique is combined with a new non-trivial estimate of \(\sum_{n\leq x}c_n^2\), and consequently the reviewer's former mean square estimate on the critical line is slightly improved. Theorem 2 also includes a sharp estimate of the error term in the mean square formula when \(\sigma\) is near 1. The author proves it by applying a mean square estimate of the error term in the asymptotic formula for \(\sum_{n\leq x}c_n\). This idea has been recently generalized and refined by S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa and the reviewer.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].