##
**Lectures on mean values of the Riemann zeta function.**
*(English)*
Zbl 0758.11036

Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. 82. Berlin: Springer-Verlag. viii, 363 p. (1991).

This book is based on lectures given by the author in the summer of 1990, at the Tata Institute. It is concerned principally with the mean values
\[
I_ k(T)=\int_ 0^ T |\zeta(1/2+it)|^{2k} dt,
\]
although results for \(\zeta(\sigma+it)\) with \(\sigma>{1\over2}\), and for intervals \([T,T+H]\) are also mentioned. After an introductory chapter there is an account of the function
\[
E(T)=I_ 1(T)-T\bigl(\log{T \over {2\pi}}+2\gamma-1\bigr),
\]
giving the \(O\)- and \(\Omega_ \pm\)-results and mean value formulae. The fourth chapter considers \(I_ k(T)\) for a general positive integer \(k\), discussing in particular the use of approximate functional equations and weighted integrals, and the evaluation of the supposed main term. For \(k\geq 3\) all this is of formal interest only, but for \(k=2\) the author gives in detail the proof of the reviewer’s fourth power moment estimate, for example. The next chapter is perhaps the highlight of the book. It describes the use of “Kloostermania” to handle \(E_ 2(T)\), the error term for \(I_ 2(T)\).

Following ideas of Motohashi, themselves still in the courses of publication, the important results \(E_ 2(T)\ll T^{2/3} \log^ C T\), \(E_ 2(T)=\Omega(T^{1/2})\), and \[ \int_ T^{T+H} E_ 2(t)^ 2 dt\;\ll\;T^{3/2} H^{3/4}\log^ C T \qquad (H\leq T) \] are established. The final chapter gives a standard account of \(I_ k(T)\) for fractional \(k\).

These lectures give a connected comprehensive account of one of the most important aspects of the theory of the Riemann zeta-function, and will be useful to all those who work in the area.

Following ideas of Motohashi, themselves still in the courses of publication, the important results \(E_ 2(T)\ll T^{2/3} \log^ C T\), \(E_ 2(T)=\Omega(T^{1/2})\), and \[ \int_ T^{T+H} E_ 2(t)^ 2 dt\;\ll\;T^{3/2} H^{3/4}\log^ C T \qquad (H\leq T) \] are established. The final chapter gives a standard account of \(I_ k(T)\) for fractional \(k\).

These lectures give a connected comprehensive account of one of the most important aspects of the theory of the Riemann zeta-function, and will be useful to all those who work in the area.

Reviewer: D.R.Heath-Brown (Oxford)