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The fourth power mean of the Riemann zeta-function. (English) Zbl 0788.11034

Bombieri, E. (ed.) et al., Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universitá di Salerno, 325-344 (1992).
This is a well-written survey, whose aim is to provide an extensive presentation of the recent developments concerning the asymptotic formula for (1) \(\int_ 0^ T |\zeta(1/2+it)|^ 4 dt\). The fundamental result is due to the author [Acta Math. 170, 181-220 (1993; Zbl 0784.11042)], who established an explicit formula for \[ I(T,\Delta)= (\Delta\sqrt\pi)^{-1} \int_{-\infty}^ \infty \bigl|\zeta( \textstyle{1\over 2}+ iT+it)\bigl|^ 4\;e^{-(t/\Delta)^ 2} dt \qquad (0<\Delta<T/\log T) \tag{2} \] by means of spectral theory of the non-Euclidean Laplacian over \(\text{SL}(2,\mathbb{Z})\). The appearance of this theory is explained by the author as follows. If quadruple sums \[ \sum_{k,\ell,m,n} f(k,\ell,m,n)= \Bigl( \sum_{k\ell=mn} + \sum_{k\ell<mn} + \sum_{k\ell>mn} \Bigr) f(k,\ell,m,n), \] which naturally appear in the treatment of (1), are regarded as sums over \(2\times 2\) integral matrices A, then one may write \[ \sum_{\text{det }{\mathbf A}>0} f({\mathbf A})= \sum_{n=1}^ \infty\;\sum_{ad=n} \sum_{b=1}^ d\;\sum_{{\mathbf A}\in\text{SL} (2,\mathbb{Z})} f\biggl(\biggl( {a\atop \phantom{d}} {b\atop d} \biggr){\mathbf A}\biggr), \] which leads to the theory of automorphic functions. The salient points in the proof of the author’s formula for \(I(T,\Delta)\) are given, together with some of the applications of (2) and a history of the results concerning (1).
The paper ends with a nice discussion of the conjectural formula for the sixth moment and N. V. Kuznetsov’s recent (unsuccessful) attempt to prove the eighth moment [Stud. Math., Tata Inst. Fundam. Research 12, 57-117 (1989; Zbl 0745.11040)].
For the entire collection see [Zbl 0772.00021].
Reviewer: A.Ivić (Beograd)

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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