*(English)*Zbl 0601.10026

This is the second edition of the text of *E. C. Titchmarsh* [“The theory of the Riemann zeta-function”, Clarendon Press, Oxford (1951; Zbl 0042.07901)] which has been of great value to generations of number- theorists. For a long time it was the only book solely on $\zeta \left(s\right)$, until the appearance of monographs of *H. M. Edwards* [“Riemann’s zeta function”, Academic Press, London (1974; Zbl 0315.10035)] and the reviewer [“Topics in recent zeta-function theory”, Publ. Math. Orsay 83.06 (1983; Zbl 0524.10032), and the expanded version “The Riemann zeta- function”, John Wiley and Sons, New York (1985; Zbl 0556.10026)].

Titchmarsh’s original has been now revised by D. R. Heath-Brown, and the present version is about 50 pages longer. This is explained in Heath-Brown’s preface to the second edition: ”Since the first edition was written, a vast amount of further work has been done. This has been covered by the end-of-chapter notes. In most cases, restrictions on space have prohibited the inclusion of full proofs, but I have tried to give an indication of the methods used whereever possible. (Proofs of quite a few of the recent results described in the end of chapter notes may be found in the book by the reviewer [“The Riemann zeta-function”, op. cit.)]. I have also corrected a number of minor errors, and made a few other small improvements to the text”.

A brief description of the contents is as follows. Chapter I, “The function $\zeta \left(s\right)$ and the Dirichlet series related to it”, is elementary. Chapter II, “The analytic character of $\zeta \left(s\right)$, and the functional equation”, provides seven proofs of the functional equation $\zeta \left(s\right)=\chi \left(s\right)\zeta (1-s)$. Chapter III. “The theorem of Hadamard and de la Vallée Poussin, and its consequences”, gives a proof of a weak form of the prime number theorem from $\zeta (1+it)\ne 0$. Chapter IV, “Approximate formulae”, discusses exponential integrals and exponential sums, and then gives the approximate functional equation for $\zeta \left(s\right)$, including the Riemann-Siegel formula.

Chapter V, “The order of $\zeta \left(s\right)$ in the critical strip”, uses the methods of Weil and van der Corput to bound $\zeta \left(s\right)$. Chapter VI, “Vinogradov’s method”, derives the bound for $\zeta (1+it)$ and the zero- free region for $\zeta \left(s\right)$ from estimates of the sum $\sum {n}^{it}$. Chapter VII, “Mean-value theorems, investigates the integrals ${\int}_{0}^{T}{\left|\zeta (\sigma +it)\right|}^{2k}\phantom{\rule{0.166667em}{0ex}}dt$ and ${\int}_{0}^{\infty}{\left|\zeta (\sigma +it)\right|}^{2k}{e}^{-\delta t}\phantom{\rule{0.166667em}{0ex}}dt$. Chapter VIII, “${\Omega}$-theorems”, contains various omega results for $\zeta (\sigma +it)$. Chapter IX, “The general distribution of zeros”, studies the functions $N\left(T\right)$, $N(\sigma ,T)$, $S\left(t\right)$ and ${S}_{1}\left(t\right)$ connected with the zeros of $\zeta \left(s\right)$. Chapter X, “The zeros on the critical line”, studies the function ${N}_{0}\left(T\right)$ and provides a proof that ${N}_{0}\left(T\right)\gg TlogT\xb7$ Chapter XI, “The general distribution of values of $\zeta \left(s\right)$ ”, discusses the values taken by $\zeta (\sigma +it)$ for various ranges of $\sigma $. Chapter XII, “Divisor problems”, is concerned with ${{\Delta}}_{k}\left(x\right)$, the error term in the asymptotic formula for the summatory function of ${d}_{k}\left(n\right)\xb7$

Chapter XIII, ”The Lindelöf hypothesis”, gives necessary and sufficient conditions for the Lindelöf hypothesis (still unsettled) $\zeta (+it)\ll {\left|t\right|}^{\epsilon}$ to hold. Chapter XIV, “Consequences of the Riemann hypothesis”, provides a detailed account of problems involving the famous Riemann hypothesis (all complex zeros of $\zeta \left(s\right)$ satisfy $\text{Re}\phantom{\rule{0.166667em}{0ex}}s=1/2)$, and shows that the Riemann hypothesis implies the Lindelöf hypothesis. This chapter is one of the highlights of Titchmarsh’s work, since to this day the Riemann hypothesis remains one of the most important unsolved problems of mathematics. Chapter XV, ”Calculations relating to the zeros”, brings some numerical calculations, such as the location of the first six zeros on $\text{Re}\phantom{\rule{0.166667em}{0ex}}s=1/2$. There are 14 pages of bibliography from the original text, plus 7 new pages added by Heath-Brown. There is no index.

The new edition of Titchmarsh’s classic, set now in paperback and reasonably priced, will be certainly welcomed by novices and experts alike.

##### MSC:

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |

11-02 | Research monographs (number theory) |

11M26 | Nonreal zeros of $\zeta \left(s\right)$ and $L(s,\chi )$; Riemann and other hypotheses |

11N37 | Asymptotic results on arithmetic functions |

11L03 | Trigonometric and exponential sums, general |