Preface for Volume One

Summary

Why have these two volumes on equivalences to the Riemann hypothesis been written? Many would say that the Riemann hypothesis (RH) is the most noteworthy problem in all of mathematics. This is not only because of its relationship to the distribution of prime numbers, the fundamental building blocks of arithmetic, but also because there exist a host of related conjectures that will be resolved if RH is proved to be true and which will be proved to be false if the converse is demonstrated. These are the RH equivalences. The lists of equivalent conjectures have continued to grow ever since the hypothesis was first enunciated, over 150 years ago.

The many attacks on RH that have been reported, the numerous failed attempts, and the efforts of the many whose work has remained obscure, have underlined the problem's singular nature. So too has its mythology. The great English number theorist, Godfrey Hardy, wrote a postcard to Harald Bohr while returning to Cambridge from Denmark in rough weather that read: “Have proof of RH. Postcard too short for proof.” He didn't believe in a God, but was certain he would not be allowed to drown with his name associated with an infamous missing proof. David Hilbert, the renowned German mathematician, was once asked, “If you were to die and be revived after five hundred years, what would you then do?” Hilbert replied that he would ask “Has someone proved the Riemann hypothesis?” More recently, towards the end of the twentieth century, Enrico Bombieri, an Italian mathematician at the Institute for Advanced Study, Princeton, issued a joke email announcing the solution of RH by a young physicist, on 1 April of course!

There are several ways in which the truth of the hypothesis has been supported but not proved. These have included increasing the finite range of values T > 0 such that the imaginary part of all complex zeros of ζ(s) up to T all have real part 1/2[68], increasing the lower bound for the proportion of zeros that are on the critical line_s = 1/ 2 [40], and increasing the size of the region in the complex plane where ζ(s) can be proved to be non-zero [63].