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Exploring the Riemann Zeta Function

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[Reviewed by
Salim Salem
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Exploring the Riemann Zeta Function is a collection of twelve articles and a preface written by Freeman Dyson. These articles turn around recent advances in the study of Riemann Zeta function. This book is meant for researchers: apart from the first and eleventh articles it goes far beyond what any undergraduate student could grasp.

The first article is an introduction to Riemann’s life and work on Zeta function. Roger Baker takes us in a good excursion through Riemann’s life and his work. He naturally focuses on the 1859 paper on prime numbers. Together with the eleventh article (on “Reading Riemann” by S. J. Patterson) it gives us a good historical picture of what influenced Riemann and the way he wrote.

In the second article, B. Berndt and A. Straub prove Ramanujan’s Formula for \(\zeta(2n+1)\) where \(n\) is an integer. They tell the story of this formula and give a correct proof of it (Ramanujan’s proof was wrong) then compare it to Euler’s formula for \(\zeta(2n)\).

In the third article, T. Cobler and M. L. Lapidus explain fractal cohomology, the result of their quest for a suitable cohomology theory to prove the Riemann Hypothesis. They replace the Frobenius operator with the derivative operator. This is used to construct an operator associated with any given meromorphic function, which in turn can be recovered using a regularized determinant involving the aforementioned operator. The authors illustrate this using rational functions, Zeta functions of algebraic curves and varieties over finite fields, and the Riemann zeta function and they prove a quantized version of Hadamard factorization theorem. cc

The fourth and fifth articles are both surveys. The first, by Friedlander and Iwaniec, is a survey of results related to exceptional characters. The second, by Goldfeld, discusses Arthur’s truncated Eisenstein series for \(SL(2,\mathbb{Z})\) and the Riemann zeta function.

Ivić, in the sixth article, discusses the cubic moments of Hardy’s function and gives some result supporting his conjecture that \(\int_1^T Z^3(t)\,dt = O_{\varepsilon} (T^{\frac34+\varepsilon})\).

In the seventh article Karabulut and Yildrim work out the correlation of Zeta zeroes with the relative maxima of the zeta function on the critical line, assuming the Riemann hypothesis. They also study the pair correlation of these maxima and the correlation of zeros of two Dirichlet \(L\)-functions.

Kowalski gives in the eighth article a proof of a version of Bagchi’s theorem and of Veronin’s universality theorem for a family of primitive cusp forms. Mossinghoff and Trudjan describe, in the ninth article, connections between oscillations in sums involving Liouville functions. They prove that \(L_0(x)>\sqrt{x}\) and \(L_1(x)<-1/\sqrt{x}\) infinitely often.

The tenth article, by Ono, Rolen and Schneider, introduces and surveys results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the twelfth and final article of the book, Rohrlich defines a Taniyama product for zeta function and proves an analogue of Merten’s theorem.

Freeman Dyson’s preface is one of the most pleasing I have ever read. He connects Riemann hypothesis to quasicrystals and he asserts his belief that someone someday will come up, using quasicrystals or other means, with a proof of Riemann hypothesis.

The best thing in this book that it contains a wide range of information which opens a lot of doors for researchers. It is good to have these formidable results in one book. I wished, however, for a table explaining the symbols and a glossary. Both would have made the reader’s life easier. Riemann’s zeta function is difficult to understand deeply, but this book is a very good help to reach that goal.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.

See the table of contents in the publisher's webpage.