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A note on the mean value of the zeta and \(L\)-functions. XI. (English) Zbl 0998.11025

Let, in standard notation, \(H_j(s) = \sum_{n\geq 1}t_j(n)n^{-s} (\operatorname{Re} s > 1)\) be the Hecke series attached to the Maass wave form \(\psi_j(z)\) corresponding to the eigenvalue \({1\over 4} + \kappa_j^2\) of the non-Euclidean Laplacian. The reviewer [J. Théor. Nombres Bordx. 13, 453-468 (2001; Zbl 0994.11020)] has shown that \[ \sum_{K\leq \kappa_j \leq K+1}\alpha_j H_j^3(\textstyle{1\over 2}) \ll K^{1+\varepsilon} \] with \(\alpha_j = |\rho_j(1)|^2/\text{ {cosh}}(\pi\kappa_j)\), where \(\rho_j(1)\) is the first Fourier coefficient of \(\psi_j(z)\). The proof depended on Y. Motohashi’s explicit formula [see his tract, ‘Spectral theory of the Riemann zeta-function’, Cambridge (1997; Zbl 0887.11001)] for the sum \(\sum_{j\geq 1}\alpha_j H_j^2(\textstyle{1\over 2})t_j(f)h(\kappa_j)\), where \(h\) is a suitable even function of fast decay. Using a similar approach M. Jutila [Number Theory, in Memory of K. Inkeri, de Gruyter, Berlin, 167-177 (2001; Zbl 0972.11041)] proved that \[ \sum_{K\leq \kappa_j \leq K+K^{1/3}}\alpha_j H_j^4(\textstyle{1\over 2}) \ll K^{{4\over 3}+\varepsilon}.(1) \] In the present paper the authors generalize (1) by proving that \[ \sum_{K\leq \kappa_j \leq K+K^{1/3}}\alpha_j |H_j^4(\textstyle{1\over 2}+it)|\ll K^{{4\over 3}+\varepsilon}(2) \] for \(0 \leq t \leq K^{1-\vartheta}, {1\over 3} < \vartheta < 1\), where the \(\ll\)-constant depends only on \(\varepsilon\) and \(\vartheta\). To prove (2), the authors first derive an approximate functional equation for \(H_j^2({1\over 2})\). The problem then becomes the estimation of \[ \sum_{m,n\geq 1}\psi(m)\psi(n)d(m)d(n)(mn)^{-1/2}(m/n)^{it} \sum_{j\geq 1}\alpha_jh(\kappa_j)t_j(m)t_j(n), \] where \(\psi\) is a suitable test-function and \(d(n)\) is the number of divisors of \(n\). The sum over \(j\) is transformed by the Kuznetsov trace formula (Motohashi, op. cit., Theorem 2.2). The ensuing sum over \(n\) is transformed by the Voronoi summation formula. After this, the problem is reduced to a triple sum whose prominent inner part is of the form \(\sum_{m\geq 1}d(m)d(m+f)U(m/f)\), with precisely defined \(U\) and the ‘shift parameter’ \(f\) lying in a suitable interval. Such sums, related to the so-called binary additive divisor problem, can be transformed by a formula of the second author [Ann. Sci. Éc. Norm. Supér. (4) 27, 529-572 (1994; Zbl 0819.11038)]. This brings back the problem to spectral theory. Thereafter delicate estimations, involving the saddle point method, the spectral large sieve and the spectral fourth moment of \(H_j({1\over 2})\) complete the proof of (2).

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F37 Forms of half-integer weight; nonholomorphic modular forms
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