On mean values of some zeta-functions in the critical strip. (English) Zbl 1050.11075

Mean values of the Riemann zeta-function \(\zeta(s)\) play a central role in zeta-function theory. Results on asymptotic formulae for the second and fourth moment are valid for \(\sigma:=\operatorname{Re} s>1/2\); however, they have been obtained by special methods which cannot be generalized to higher moments. The \(2k\)-th moment of \(\zeta(s)\) can be written as \[ \int_1^T|\zeta(\sigma+it)|^{2k} \,dt=T\sum_{n=1}^\infty{d_k^2(n)\over n^{2\sigma}}+R(k,\sigma;T),\quad R(k,\sigma;T)=o(T), \] where \(1/2<\sigma_0(k)\leq\sigma\leq 1,T\to\infty\), and \(d_k(n)\) denotes the \(n\)th coefficient in the Dirichlet series expansion of \(\zeta(s)^k\).
In the paper under review several new bounds for \(R(k,\sigma;T)\) are obtained. Let \[ \beta_k=\inf\left\{b_k\geq 0: \int_1^x\Delta_k^2(y) \,dy\ll x^{1+2b_k}\right\}, \] where \(\Delta_k\) is the error term in the asymptotic formula for the summatory function of \(d_k(n)\). The author proves that, for fixed \(\sigma\) satisfying \(\max\{\beta_k,1/2\}<\sigma<1\) and every fixed integer \(k\geq 3\), \[ R(k,\sigma;T)\ll_\varepsilon T^{{2(1-\sigma)\over 1-\beta_k}+\varepsilon}. \] This improves a recent result of S. Kanemitsu et al. [Monatsh. Math. 136, No.1, 17-34 (2002; Zbl 1022.11047)] (it should be noted that their result holds for a general class of Dirichlet series possessing a functional equation with multiple gamma-factors; however, the present approach applies also to these Dirichlet series).
Furthermore, let \(\sigma_k^*\) be the infimum of \(\sigma^*\geq 1/2\) such that \[ \int_1^T|\zeta(\sigma^*+it)|^{2k} \,dt\ll_\varepsilon T^{1+\varepsilon} \] for any \(\varepsilon>0\). Then the author shows that, for fixed \(\sigma\) satisfying \(\sigma_k^*<\sigma<1\) and every fixed integer \(k\geq 3\), \[ R(k,\sigma;T)\ll_\varepsilon T^{{2(1-\sigma)\over 2-\sigma_k^*-\sigma}+\varepsilon}. \] This improves an old result of the author [The Riemann zeta-function. Wiley, New York (1985; Zbl 0556.10026)] in the whole range for \(\sigma\).
These improvements are remarkable and the method of proof is rather simple. The expressions in question are directly integrated, which leads to sharper estimates than those that would have resulted from the application of the Cauchy-Schwarz inequality.
Finally, the author applies his method to other Dirichlet series. For instance, he obtains an asymptotic formula for the mean-square of Rankin-Selberg \(L\)-functions, which improves a result of K. Matsumoto [Number theory, Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri (2001; Zbl 0972.11075)] in the range \({3\over 4}<\sigma<1\).


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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