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Spectral mean values of Maass waveform $$L$$-functions. (English) Zbl 0759.11026
This paper brings forth several results on spectral mean values of Maass waveform $$L$$-functions $$H_ j(s)$$ (Hecke series). Such results have recently found important applications to the fourth moment of $$\left|\zeta\left({1\over 2}+it\right)\right|$$. Major advances in this field have been made by the author [Proc. Japan Acad., Ser. A 65, 143-146 (1989; Zbl 0684.10036), ibid. 273-275 (1989; Zbl 0699.10055), “An explicit formula for the fourth power mean of the Riemann zeta- function”, Acta Math. (in press)]. By using the author’s explicit formulas and results on spectral mean values of this paper the author and the reviewer have proved [Proc. Japan Acad., Ser. A 66, 150-152 (1990; Zbl 0688.10037), “On the fourth power moment of the Riemann zeta- function”, submitted to J. Number Theory, “The mean square of the error term for the fourth moment of the zeta-function” (to appear)] $E_ 2(T)=O(T^{2/3}\log^ cT),\quad E_ 2(T)=\Omega(T^{1/2}),\int^ T_ 2E^ 2_ 2(t) dt=O(T^ 2\log^ cT),$ where $$E_ 2(T)$$ is the error term in the asymptotic formula for $$\int^ T_ 0\left|\zeta\left({1\over 2}+it\right)\right|^ 4 dt$$.
The author starts by considering the expression $I(u,v;f;h):=\sum^ \infty_{j=1}\varepsilon_ j\alpha_ jt_ j(f)H_ j(u)H_ j(v)h(\kappa_ j),$ where $$f\in\mathbb{N}$$, $$h(r)$$ is an even entire function of exponential decay such that $$h(\pm{i\over 2})=0$$, $$\varepsilon_ j$$ is the parity sign of the Maass form to which the Hecke series $$H_ j(s)$$ is attached, and $$\alpha_ j=|\rho_ j(1)|^ 2/ch(\pi\kappa_ j)$$ in standard notation. This is an entire function of $$u$$ and $$v$$ to which the trace formula of Kuznetsov is applied with rigour and precision. A spectral decomposition of $$I(u,v;f;h)$$ is obtained, which permits the author to deduce from it several interesting results. Here we shall mention only $\sum_{\kappa_ j\leq K}\alpha_ jH^ 4_ j({1\over 2})\ll K^ 2\log^{20}K$ and $\sum_{\kappa_ j\leq G}\alpha_ jH^ 2_ j\left({1\over 2}\right)=2\left({G\over\pi}\right)^ 2(\log G+\gamma- {1\over 2}-\log(2\pi))+O(G\log^ 6G). \tag{1}$ Note that (1) corrects and rigorously establishes a formula of N. V. Kuznetsov [Th. 11 in Number Theory and related topics, Pap. Ramanujan Colloq., Bombay/India 1988, Stud. Math., Tata Inst. Fundam. Res. 12, 57-117 (1989; Zbl 0745.11040)].
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations