Mean values of certain zeta-functions on the critical line
Let \(f(s)\) be the Riemann zeta-function \(\zeta(s)\), or a Dirichlet L- function \(L(s,\chi)\) with a principal character \(\chi\) (mod q), or a Dedekind zeta-function \(\zeta_ K(s)\), or a zeta-function \(\phi(s+(k- 1)/2)\) of a cusp form of weight k on the full modular group. The paper proves that \[ (*)\;\int^{T}_{0}| f(+it)|^{\lambda} dt \sim T\;(T\to \infty) \] for suitable \(\lambda =\lambda (T)\geq 0\). Specifically, if \(\psi\) (T)\(\to \infty\) with T, we may take any \(\lambda \leq (\psi (T)\log \log T)^{-1/2}\) for \(\zeta\) (s) and L(s,\(\chi\)), uniformly for \(q\leq T^{1-\epsilon}.\) Moreover it is shown that for these two functions the asymptotic relation (*) fails for \(\lambda =C(\log \log T)^{-1/2}\) if C is a sufficiently large constant. For \(\phi (s+(k-1)/2)\) it is proved that (*) holds for \(\lambda\leq (\psi (T)\log \log T)^{-1}\), and similarly for \(\zeta_ K(s)\) with K quadratic. However for higher degree fields only the smaller range \(\lambda\leq (\psi(T)\log T)^{-1}\) is handled. The methods used are based on \textit{M. Jutila}'s uniform version [Bull. Lond. Math. Soc. 15, 513-518 (1983; Zbl 0493.10043)] of the reviewer's work [J. Lond. Math. Soc., II. Ser. 24, 65-78 (1981; Zbl 0431.10024)] on fractional moments of \(\zeta (+it)\).
scientific article; zbMATH DE number 4158745