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The above comment appears in Riemann’s memoir to the Berlin Academy of Sciences (Section 12.2). It seems to be a passing thought, yet it has become, arguably, the most central problem in modern mathematics.

In this chapter we develop some of the more important, and beautiful, results in the classical theory of the zeta function. The material is mathematically sophisticated; however, our presentation should be accessible to the reader with a first course in complex analysis. At the very least, the results should be meaningful even if the details are elusive.

The Riemann hypothesis has been open since its “proposal” in 1859. The most direct form of attack is to search for a counterexample, and also to search computationally for insight. Mathematicians from the time of Riemann have developed a large body of computational techniques and evidence in support of the Riemann hypothesis. The advent of modern computing empowered mathematicians with new tools, and motivated the development of more efficient algorithms for the computation of ζ(s).

The Riemann hypothesis has endured for more than a century as a widely believed conjecture. There are many reasons why it has endured, and captured the imagination of mathematicians worldwide. In this chapter we will explore the most direct form of evidence for the Riemann hypothesis: empirical evidence. Arguments for the Riemann hypothesis often include its widespread ramifications and appeals to mathematical beauty; however, we also have a large corpus of hard facts. With the advent of powerful computational tools over the last century, mathematicians have increasingly turned to computational evidence to support conjectures, and the Riemann hypothesis is no exception.

In this chapter we discuss several statements that are equivalent to the Riemann hypothesis. By restating the Riemann hypothesis in different language, and in entirely different disciplines, we gain more possible avenues of attack. We group these equivalences into three categories: equivalences that are entirely number-theoretic, equivalences that are closely related to the analytic properties of the zeta function and other functions, and equivalences that are truly cross-disciplinary. These equivalences range from old to relatively new, from central to arcane, and from deceptively simple to staggeringly complex.

It is common in mathematics to generalize hard problems to even more difficult ones. Sometimes this is productive and sometimes not. However, in the Riemann hypothesis case this is a valuable enterprise that allows us to see a larger picture. These stronger forms of the Riemann hypothesis allow the proof of many conditional results that appear to be plausible (see Sections 11.1 and 11.2) and add to the heuristic evidence for the Riemann hypothesis.

The consequences of a proof of the Riemann hypothesis in elementary number theory are apparent, as are the connections to applications such as cryptography. Aside from these considerations, a large body of theory has been built on the Riemann hypothesis. In this chapter we consider some of the more important statements that become true should a proof of the Riemann hypothesis be found.

In this chapter we discuss four famous failed attempts at the Riemann hypothesis. Though flawed, all of these attempts spurred further research into the behavior of the Riemann zeta function.

The following is a reference list of useful formulas involving ζ(s) and its relatives. Throughout, the variable is assumed to lie in N.

There is almost three hundred years of history surrounding the Riemann hypothesis and the prime number theorem. While the authoritative history of these ideas has yet to appear, this timeline briefly summarizes the high and low points.