The twelfth moment of central values of Hecke series
The author proves the bound \[ \sum_{\kappa_j\leq K}\alpha_j H^{12}_j({{1\over2}}) \ll_\varepsilon K^{4+\varepsilon},\leqno(1) \] which may be viewed as the analogue of \textit{D. R. Heath-Brown}'s estimate [Q. J. Math., Oxf. (2) 29, 443--462 (1978; Zbl 0394.10020)] \[ \int_0^T| \zeta({\textstyle{1\over2}}+it)| ^{12}\text{ d}t \ll_\varepsilon T^{2+\varepsilon}\leqno(2) \] for the Riemann zeta-function \(\zeta(s)\). In fact, in the proof of (1) the author uses (2). As usual, \(H_j(s)\) is the Hecke series attached to the Maass wave form for the \(j\)-th eigenvalue \({1\over4}+\kappa_j^2\) of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. Also \(\alpha_j = | \rho_j(1)| ^2/\cosh(\pi\kappa_j)\), where \(\rho_j(1)\) is the first Fourier coefficient of the Maass wave form to which \(H_j(s)\) is attached. The present work is related to the author's earlier work [Proc. Turku Symp. Number Theory, de Gruyter, Berlin, 167--177 (2001; Zbl 0972.11041)] which contained the bound \[ \sum_{| \kappa_j-K| \leq K^{1/3}}\alpha_j H^{4}_j({\textstyle{1\over2}}) \ll_\varepsilon K^{4/3+\varepsilon}, \] which like (1) implies, in view of \(\alpha_j \gg_\varepsilon \kappa_j^{-\varepsilon}\), the (strongest known) ``subconvexity'' bound \[ H_j({\textstyle{1\over2}}) \ll_\varepsilon \kappa_j^{1/3+\varepsilon}. \leqno(3) \] The estimate (3) was previously proved by the reviewer [J. Théor. Nombres Bordx. 13, 554--568 (2001; Zbl 0994.11020)] as a consequence of his result \[ \sum_{| \kappa_j-K| \leq 1}\alpha_j H^{3}_j({\textstyle{1\over2}}) \ll_\varepsilon K^{1+\varepsilon} \] and the non-negativity of \(H_j({\textstyle{1\over2}})\). To prove (1), the author proves first that \[ \int_K^{2K} F^2(x)\,\text{ d}x \ll_\varepsilon K^{3+\varepsilon}, \quad F(x) = \sum_{x<\kappa_j\leq x+1} \alpha_j H^{4}_j({\textstyle{1\over2}}), \] and a related bound for the integral where \(F(x) \geq U\). The proofs use several techniques from spectral theory of the non-Euclidean Laplacian.
scientific article; zbMATH DE number 2106058
scientific article