New analytic problems over imaginary quadratic number fields (Q2710130)

The main objective of this paper is to obtain a spectral decomposition of NEWLINE\[NEWLINE \int_{-\infty}^\infty |\zeta_k({\textstyle{1\over 2}}+it)|^4g(t) dt, \tag{1}NEWLINE\]NEWLINE where \(k = {\mathbb Q}(i)\) is the Gaussian number field with the corresponding zeta-function NEWLINE\[NEWLINE \zeta_k(s) = {\textstyle{1\over 4}}\sum_{n\not=0}|n|^{-2s} \qquad(\Re s > 1), NEWLINE\]NEWLINE and \(g(t)\) is assumed to be real on \({\mathbb R}\), and holomorphic and of rapid decay in any fixed horizontal strip. The corresponding (less difficult problem) when in the above integral one has the classical Riemann zeta-function \(\zeta(s)\) instead of \(\zeta_k(s)\) has been successfully resolved by the author [see e.g., his monograph ``Spectral theory of the Riemann zeta-function'', Cambridge Univ. Press, Cambridge (1997; Zbl 0878.11001)]. Since \(\zeta_k(s) = \zeta(s)L(s)\) where NEWLINE\[NEWLINE L(s) = \sum_{n=0}^\infty (-1)^n(2n+1)^{-s}\qquad(\Re s > 1), NEWLINE\]NEWLINE the problem corresponds to the evaluation of NEWLINE\[NEWLINE \int_{-\infty}^\infty |\zeta({\textstyle{1\over 2}}+it)|^8g(t) dt, NEWLINE\]NEWLINE although there are naturally important differences, which are pointed out in the text. After the initial reduction, Ramanujan sums over \(k\) are introduced. The functional equation for NEWLINE\[NEWLINE \zeta_k(s;\nu,\xi) := \sum_{n+\xi\not=0}((n+\xi)/|n+\xi|)^\nu|n+\xi|^{-2s} \qquad(\Re s > 1), NEWLINE\]NEWLINE with arbitrary \(\nu \in {\mathbb Z}\) and \(\xi \in {\mathbb C}\) is derived. In providing a necessary analytic continuation the author needs several integrals containing Bessel functions, which are given in five lemmas. Sums of Kloosterman sums over \(k\) naturally emerge, and the corresponding trace formula (Theorem 2) plays an essential part and is certainly of intrinsic interest as is the binary additive divisor problem over \(k\). The paper ends with a discussion of the analytic problems encountered in the author's present work. Although not yet complete, the present work is an important step in attaining the author's objective, namely to provide a full spectral decomposition of (1). NEWLINENEWLINE\noindent Reviewer's remark: The problem of the fourth power moment of the Dedekind zeta-function has been resolved recently by \textit{R. Bruggeman} and the author in their submitted article ``Sum formula for the Kloosterman sums and the fourth moment of the Dedekind zeta-function over the Gaussian number field'' (see also the announcement ``A note on the mean value of the zeta and \(L\)-functions. X'' by the same authors, which is in print in Proc. Japan Acad.).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].