Reassessing Riemann's Paper On the Number of Primes Less Than a Given Magnitude

In his only paper on the theory of numbers, Riemann established a deep analytic method to study the distribution of the prime numbers. The key to that method is connecting primes numbers to the complex function \(\zeta(s)=\sum_{n=1}^{\infty}n^{-s}\), which today is known as the Riemann zeta function. In his paper, Riemann obtained an analytic continuation and a functional equation for \(\zeta(s)\), and further made five important conjectures that showed that there is a deep connection between the distribution of the prime numbers and the zeros of \(\zeta(s)\).

Because of the huge effect of this paper on mathematicians studying the primes, several books have been written about Riemann zeta function, such as Titchmarsh’s The theory of the Riemann zeta-function and Ivić’s The Riemann zeta-function. Theory and applications. Riemann’s paper itself was explored in depth by Edwards in Riemann’s zeta function.

The book under review, as its title indicates, elaborates on Riemann’s paper, starting with a short biography of Riemann and an account of the scientific atmosphere of his time. In Chapter 1 the author considers Euler’s product formula and a simple extension of \(\zeta(s)\) to the half-plane \(\Re(s)>0\). In Chapters 2–4, the author studies Riemann’s formula for the prime counting function, discusses Mellin and Fourier transforms, and explains Riemann’s method of analytic continuation of \(\zeta(s)\). These are all results proved in Riemann’s paper.

Riemann also made some important conjectures, which are studied in following chapters. In Chapter 5, the author studies the Hadamard product for \(\zeta(s)\), in Chapter 6 he studies the “explicit formula,” which provides a neat connection between prime counting function and the zeros of \(\zeta(s)\), and in Chapter 7 he studies the number of non-trivial zeros. Finally, in Chapter 8, the author introduces the use of the zeta function in connection with regularizing certain problems in quantum physics where infinities occur.

The book is very brief, but nonetheless at some points the notations are not consistent. A meticulous reader should be careful about this and also about some incorrect expressions.

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.