The fourth moment of central values of Hecke series (Q2710123)

Let \(H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}\) denote the Hecke series attached to the eigenvalue \(\lambda_j = \kappa_j^2 + 1/4\) of the hyperbolic Laplacian, and let as usual NEWLINE\[NEWLINE \alpha_j = {|\rho_j(1)|^2\over \cosh(\pi\kappa_j)},\quad NEWLINE\]NEWLINE where \(\rho_j(n)\) is the \(n\)-th Fourier coefficient corresponding to the Maaß wave form which generates \(H_j(s)\). The reviewer [``On sums of Hecke series in short intervals'', J. Théor. Nombres Bordx. (2001), in print] proved that NEWLINE\[NEWLINE \sum_{|\kappa_j-K|\leq 1}\alpha_j H_j^3({\textstyle{1\over 2}}) \ll_\varepsilon K^{1+\varepsilon}.\leqno(1) NEWLINE\]NEWLINE In view of the nonnegativity of \(H_j({\textstyle{1\over 2}})\) and \(\alpha_j \gg \kappa_j^{-\varepsilon}\) this yields NEWLINE\[NEWLINE H_j({\textstyle{1\over 2}}) \ll_\varepsilon \kappa_j^{{1\over 3}+\varepsilon}, \leqno(2) NEWLINE\]NEWLINE which is hitherto the best bound of its kind. Now the author proves the bound NEWLINE\[NEWLINE \sum_{|\kappa_j-K|\leq K^{1/3}}\alpha_j H_j^4({\textstyle{1\over 2}}) \ll_\varepsilon K^{{4\over 3}+\varepsilon}.\leqno(3) NEWLINE\]NEWLINE The bounds (1) and (3) are independent of one another; note that (3) also yields (2), only \(H_j({\textstyle{1\over 2}}) \geq 0\) is not needed this time. Both the proof of (1) and (3) depend on the important formula of \textit{Y. Motohashi} [see e.g. his monograph ``Spectral theory of the Riemann zeta-function'', Cambridge Univ. Press (1997; Zbl 0878.11001)] for the transformation of the sum NEWLINE\[NEWLINE \sum_{j=1}^\infty \alpha_j H_j^2({\textstyle{1\over 2}})t_j(f)h_(\kappa_j), \leqno(4) NEWLINE\]NEWLINE where \(f\) is a positive integer and the entire function \(h\) satisfies certain conditions. To be able to apply (4), the author derives first an approximate functional equation for \(H_j^2({\textstyle{1\over 2}})\), expressing it in terms of two Dirichlet polynomials of length \(3K^2\). By using the multiplicativity relation NEWLINE\[NEWLINE t_j(m)t_j(n) = \sum_{\delta|m,\delta|n}t_j\left({mn\over\delta^2}\right) NEWLINE\]NEWLINE the smoothened sum in (3) is transformed to a sum with only one \(t_j\) function weighted by the number of divisors function \(d(n)\). After the formula for the transformation of (4) is applied, one obtains a sum to which Motohashi's formula for NEWLINE\[NEWLINE \sum_{f=1}^\infty d(f)d(f+m)W(f/m) NEWLINE\]NEWLINE may be applied, where \(W\in C^\infty\) is a function with compact support on the positive real axis. The proof is completed then by a careful analysis of the ensuing expression and an application of the spectral large sieve inequality [see Motohashi, op. cit.].NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].