Spectral mean values of Maass waveform \(L\)-functions.

*(English)*Zbl 0759.11026This 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)].

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)].

Reviewer: A.Ivić (Beograd)

##### MSC:

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 |

##### Keywords:

Riemann zeta-function; Kuznetsov trace formula; fourth power mean; Maass waveform \(L\)-functions; Hecke series; spectral mean values; error term; asymptotic formula
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\textit{Y. Motohashi}, J. Number Theory 42, No. 3, 258--284 (1992; Zbl 0759.11026)

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##### References:

[1] | Ivić, A, () |

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[5] | Kuznetsov, N.V, Petersson hypothesis for forms of weight zero and Linnik’s conjecture: sums of Kloosterman sums, Math. USSR-sb., 39, 299-342, (1981) · Zbl 0461.10017 |

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[10] | Takhtajan, L.A; Vinogradov, A.I, An estimate of the residuum of Rankin’s L-series, Dokl. akad. nauk USSR, 267, 30-34, (1982), [In Russian] |

[11] | Whittaker, E.T; Watson, G.N, () |

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