Lectures on the mean-value and Omega-theorems for the Riemann Zeta-function.

*(English)*Zbl 0845.11003Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. 85. Berlin: Springer-Verlag. Bombay: Tata Institute of Fundamental Research, xiii, 169 p. DM 44.00; öS 321.20; sFr 44.00 (1995).

This book presents a connected account of the author’s many works on upper and lower bounds for

$${\int}_{T}^{T+H}{\left|\zeta (\sigma +it)\right|}^{2k}dt$$

and related integrals. Although most of these results have therefore appeared before, it is convenient to have them collected together, with the proofs based on common principles, which are established here in the first two chapters.

The statements of many of the theorems are long and involved. As two pithy examples we mention: (1) Let $z$ be a complex number of modulus 1, and let $cloglogT\le H\le T$ with a sufficiently large constant $c$. Then

$${max}_{T\le t\le T+H}\left|\zeta {(\frac{1}{2}+it)}^{z}\right|\ge exp\left(\frac{3}{4}\sqrt{\frac{logy}{loglogy}}\right)\xb7$$

(2) If $H={T}^{\frac{1}{2}+\epsilon}$ with $\epsilon >0$ constant, then

$$\int}_{T}^{T+H}{\left|\zeta (\frac{1}{2}+it)\right|}^{2/k}dt{\ll}_{\epsilon}H{(logT)}^{1/{k}^{2}$$

for any positive integer $k$.

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