In his 1859 memoir “On the number of prime numbers less than a given quantity,” Riemann showed that the zeta-function has a key role in studying the distribution of prime numbers. Among several other assertions, he made an explicit connection between the distribution of the prime numbers and the distribution of the zeros of the Riemann zeta-function. This connection motivated many mathematicians to study those zeros from several points of view.

The book under review presents a number of essential research directions in this area in a friendly fashion. It focuses mainly on two questions: the location of zeros in the critical strip (the strip on the complex plane consisting of numbers with real part between 0 and 1) and the proportion of zeros on the critical line (the line at the center of the strip, consisting of numbers with real part 1/2).

The book consists of two main parts. The first part looks at Riemann’s work on the zeta-function the classical work that followed, mostly aimed at proving his conjectures and applying the results to study the distribution of the primes. The second part focuses on the Riemann’s last and unsolved conjecture, which is the well-known Riemann hypothesis, asserting that all non-real zeros of the Riemann zeta-function are located on the critical line. The author describes the work in this direction initiated by Hardy and Littlewood (who proved that there are infinitely many zeros on the critical line) and then improved remarkably by Selberg (who showed that a positive proportion of the zeros rest on the critical line). Explicit improvements on the method of Selberg due to Levinson (who proved 1/3 of the zeros rest on the critical line) and to Conrey (who proved 2/5 of the zeros rest on the critical line) are explained as well. Finally, the author presents his own improvement, asserting that

The Riemann hypothesis is more likely to be true than not!

This is one of the first books to present Levinson’s argument. The book is quite technical, and readers need a basic knowledge in complex function theory and also analytic number theory to follow the details. The chapters are short, well-motivated, and well written; there are several exercises. Thus, the book can serve as a source for researchers working on the Riemann zeta-function and also to be a good text for an advanced graduate course.

Buy Now

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.