1 - Introduction

Summary

Chapter Summary

This chapter is discursive. It begins with an overview of the early history of the Riemann hypothesis (RH) and the evolution of ideas relating to the Ramanujan–Robin inequality. Then in Section 1.3 there is a summary of the contents of the entire volume, first in brief and then in more detail. The section also describes the tables in Appendix A and the software RHpack in Appendix B.

There is a section on notational conventions and special notations, most of which are quite standard. A guide to the reader and two problems complete the chapter.

Early History

Here the main players in the evolution of the Riemann hypothesis are noted: Euclid, Euler, Gauss, Dirichlet and, last but not least, Riemann himself. The first is Euclid of Alexandria (Figure 1.1) who lived around 300 BCE. His Elements includes a proof that there are an infinite number of primes, and that they are the fundamental building blocks of numbers, through the unique factorization of integers. How are the primes distributed? On the face of it, the only pattern appears to be that all are odd, except 2. Do they appear completely at random, or are they uniformly distributed in some sense?

Don Zagier gives a good description of this random/uniform dichotomy: “The first fact is the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.”

“The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.”

Euler (1707–1783; Figure 1.2) was an amazing mathematician, who wrote a text on calculus, including the first treatment of trigonometric functions, and invented many parts of mathematics that are important today, including the calculus of variations, graph theory and divergent series. In number theory, quadratic reciprocity and Euler products are two of his many contributions, as well as his extensive work on the zeta function. In fact, Euler gave birth to the Riemann zeta function, writing down its definition for the first time.