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Riemann's Zeta Function: Harold M. Edwards

Superb study of one of the most influential classics in mathematics examines the landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics.

Review:

A good guide to Riemann, the prime number theorem, and the Riemann hypothesis

Chapter 1 analyses Riemann's paper in detail. The zeta function is the product over all primes of 1/(1-1/p^s). Taking the logarithm, one obtains an expression involving the density of primes. So to say something about the density of primes one must say something about the log of the zeta function. Riemann does this by allowing the variable s of the zeta function to be complex, which enables him to prove the functional equation of the zeta function and the product representation of the xi function defined through it. From here he can derive an expression for log zeta, thus yielding an expression for prime density. Since it comes from log zeta, this expression depends on the poles of log zeta, i.e. the zeros of the zeta function. Riemann feels that all nontrivial zeros have real part 1/2, but this doesn't really matter left now since the term in the prime density expression depending on the zeros is "periodic" in any case and Riemann thus discards it without much harm when he derives his expression for the number of primes less than x. Hadamard and von Mangoldt later gave more rigourous proofs of the product formula for the xi function (chapter 2) and Riemann's prime density expression (chapter 3). As indicated by Riemann, further progress depends on an understanding of the zeros of the zeta function. Indeed, in this way Hadamard proved the prime number theorem (chapter 4), i.e. that the prime counting function is asymptotically equal to the logarithmic integral, and also in this way de la Vallee Poussin derived a bound for the error in this approximation (chapter 5). The Riemann hypothesis would imply a better bound. Chapters 6,7,8,9,11 deal with the pitiful progress towards the Riemann hypothesis, including computational aspects. Chapter 10 tries to hot up the Fourier analysis used in the classical works by putting it in terms of self-adjoint operators and so on. Chapter 12 "Miscellany" includes a proof of the prime number theorem that "is 'elementary' in the technical sense", but, as Edwards admits, it is neither straightforward, nor natural, nor insightful.

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