In Pursuit of Zeta-3

The Riemann zeta function \( \zeta(s) \) is introduced, first for (positive) integer values \( s \) as the infinite sum of the reciprocals of the \( s \)th power of the positive integers. It is remarkable that simple expressions of the form \( \zeta(2n) = \pi^{2n} /c_{2n} \) , with \( c_{2n} \) some integer, can be found for all positive integers \( n \). However, for any odd argument \( s = 2n + 1 \) with \( n > 0\), many relations, integrals, and series expressions are known, but no simple expression of a form  similar to the even case is known. Obviously, \( \zeta(1) \) is the harmonic series and thus divergent. The Apéry constant \( \zeta(3) \) is the first and supposedly the simplest problem for which no solution has yet been found. Hence the title of the book. The definition of \( \zeta(s) \) for complex values is not formally given since it is not the focus here, but \( s \) is tacitly understood to be complex in certain integral expressions.

 

Roger Apéry’s name is associated with \( \zeta(3) \) because he proved its irrationality. Euler however, is the main character of this book, and Nahin is obviously an admirer of his work. He even includes a picture of Euler with the caption "The hero of this book". Euler connected the reciprocal of \( \zeta(s) \) to the product of \( 1-p^{-s} \) over all prime numbers p. This implies that all the negative integers are ‘trivial’ zeros of \( \zeta(s) \). This immediately links up with the famous Riemann Hypothesis (RH) and explains why \( \zeta \) is called the Riemann-zeta function. The RH states that all the non-trivial zeros of \( \zeta \) (i.e., the ones different from the negative integers) are on a vertical line in the complex plane through \( (1/2, 0) \). This is one of the Millennium Prize Problems.  Solving it is worth a million dollars. Nahin, however, still considers the expression for the Apéry constant to be more interesting than the RH because his criterion to deserve the title of "most puzzling mathematical problem" is that it should be an easy to understand but challenging and unsolved problem that has some connection to the real physical world. Clearly the RH requires complex analysis to properly understand the problem.  Nevertheless, skipping details, the RH is introduced in this book because it concerns the zeta function.

 

Paul Nahin is an electrical engineer who has written some 20 books on popular mathematics and physics, including biographies of Boole, Heaviside, and Shannon, as well as one about tricks to solve all kinds of integrals. The latter are frequently applied in the many computations involving series and integrals in this book. The mathematics used is not always applications of standard facts taught in a calculus course, but often consists of a sequence of successive substitutions. They are very tricky and a reader would probably

not have found these on her own, but they eventually always lead to the desired formula. Every step is clearly explained, which can result in lengthy sequences of formulas that sometimes continue for several pages in a row. In some places, the manipulations need finer mathematical justification, especially when mixing infinite processes like limits, integrals, and series. This may cause some teeth grinding for mathematical purists, but this is the price one has to pay for avoiding too much ballast. Some warnings are given, but the reader is guaranteed that it is justified, and there are at many places numerical checks (using MATLAB – some code is included) to verify the results.  The target readership is thus clearly all bright young students at the calculus level. They should be able to follow the arguments relatively easily with few exceptions and they should be able to solve the challenge problems that are scattered throughout the text (solutions are given at the end), although some of the problems are true challenges.

 

The contents of the book include convergence of infinite series, the Gauss, zeta, gamma, digamma, and beta functions, and all kinds of expressions relating infinite series and integrals. Euler clearly plays a central role throughout the book, but many other historical contributors are mentioned (Gauss, the Bernoullis, Ramanujan,...) since the results are situated in their historical context. In the second half of the book Nahin is working towards the RH conjecture and he starts by introducing Fourier series and the Fourier trans-

form. The transition from a Fourier series for a periodic function to the Fourier transform of a non-periodic function is not easily explained on an intuitive basis, but Nahin makes a commendable effort. The Fourier analysis avoids complex analysis as much as is possible given the context. Formulas like Poisson’s and Parseval’s are derived only for what they are needed, thus not placing them in a context of vector or Hilbert spaces, orthogonal basis, etc., but following the long way of integrating and manipulating series. There are

some short excursions and side tracks mentioning other popular items like the four-color problem, Fermat’s last theorem, fractals, the Collatz conjecture, etc.

 

Nahin’s style is entertaining, directly addressing his readers, and making their thoughts explicit like "...double sum is zero. Is it? Yes. Why? Think about this for a while, then I’ll show you (in a box at the end of this section) a simple plausible argument..." (p.225). This trick of announcing a postponed result is used several times to keep you reading. For example, it takes at the beginning of the book several forward references and many pages before the zeta-3 problem is finally revealed. Some statements may need some explanation which is often given in one of the many footnotes.  Other important issues are isolated in special boxes like the one announced in the above quote. In my opinion the formulas are a bit over the top.  Does it really need the be explained that \( (−1)^{k} (-1)^{k}=(−1)^{2k} = 1\)? This level of detail makes many long computations come across as a bit boring if the reader is skilled in calculus, although I can imagine that it is needed for younger readers for whom this material is all new and who are struggling to keep up in this world of unfamiliar functions, series and integrals. There are plenty of occasions for the reader to test her skills with the exercises.

 

This is a highly recommended book for interested students. It is a marvelous introduction to Fourier analysis and to all these functions and relations so near to the problem stated in the Riemann Hypothesis and yet using only basic calculus. Not all the techniques are standard and some tricks are far-fetched, but it is certainly an excellent invitation to train your calculus skills. I could only spot one typo: on page 88: "and = r sin(θ)" (the y = is missing). With so many formulas, it would be a miracle if there were not others. I am sure the most complicated formulas were triple checked or more, but, as shown above, they can be overlooked at unexpected places.  Let it be another challenge for the reader to spot another one. It will keep you focused.

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Adhemar Bultheel is emeritus professor at the Department of Computer Science of the KU Leuven (Belgium). He has been teaching mainly undergraduate courses in analysis, algebra, and numerical mathematics. More information can be found on his homepage https://people.cs.kuleuven.be/~adhemar.bultheel/