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Equivalents of The Riemann Hypothesis, Volume Two: Analytic Equivalents

Kevin Broughan
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and its Applications 165
[Reviewed by
Steven Deckelman
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The Riemann Hypothesis has aptly been described as the holy grail of mathematics. Like a medieval dragon that smote all heros who dared try slaying it, the proof or disproof of the Riemann Hypothesis has consistently withstood the efforts of the best mathematical minds over the past century and a half. This includes, incidentally, Riemann himself, who neither offered a proof nor gave any indication he had one.

The Riemann Hypothesis is a remarkable assertion about two seemingly very disparate mathematical objects, the distribution of prime numbers and the location of zeros of a specific analytic (actually meromorphic) function. Originally conjectured by Riemann in 1859 in a short note to the Prussian Academy that contained no proofs, this problem has proved to be a particulary singular and recalcitrant adversary. Mathematical problems like this are important not only in themselves and the problems they would settle upon their resolution, but also because of the mathematics that is created through the attempts of many mathematicians over the years to understand them. One way to see this is to examine the myriad techniques that have been developed connecting it to other areas of mathematics through its various equivalents. In the case of the Riemann Hypothesis the connections are astonishing and include such diverse areas as number theory (of course), numerical analysis, graph theory, group theory, matrix theory, functional analysis (Banach and Hilbert spaces), orthogonal polynomials, Hermitian forms, discrete measures and even quantum mechanics (Hilbert-Pólya conjecture).

There are many good books on the Riemann hypothesis, so this review will not dwell on the origins of the conjecture except to say that its connection to the distribution of primes first arises in the Euler product formula \[ \zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p} (1-p^{-s})^{-1}, \] initially defined for \(s\in\mathbb{C}\) in the half-plane \(Re(s)>1\) but then extended by analytic continuation on the whole complex plane except at \(s=1\) where it has a simple pole of residue \(1\). The existence of the pole at \(s=1\) implies that the number of primes are infinite. It is easy to see that \(\zeta\) has simple zeros at the negative even integers and has been known for some time (proved first by Hardy) that it has infinitely many zeros on the critical line \(\textrm{Re} (s)=\frac{1}{2}\). The Riemann Hypothesis is, of course, that apart from the trivial zeros at the negative even integers, all of its nontrivial zeros fall on this critical line and no nontrivial zero off this critical line has ever been found.

More generally there are various generalizations such as the “Riemann Hypothesis” for Dirichlet L-functions \[ L(s,\chi)=\sum\frac{\chi(n)}{n^s}\] where \(\chi\) is a Dirichlet character. The classical zeta function corresponds to \(\chi(n)=1\).

This two volume catalogue of many of the various equivalents of the Riemann Hypothesis by Kevin Broughan is a valuable addition to the literature. Its intended audience include graduate students and researchers in number theory, though most of it is quite accessible to non-specialists, the main prerequisite being a graduate course in analysis, especially complex analysis. The important ideas are summarized in several appendices. But even some of the more advanced topics, e.g. the Weil conjectures in chapter 9 of Volume Two would be good starting points for those wanting to learn more about these topics.

Volume One begins with a chapter on the history of the Riemann Hypothesis and its derivation and is mostly classical in flavor. It is mostly concerned with various equivalents that have been given involving arithmetic functions, for example the prime counting function \[\pi(x)=\sum_{p\leq x} 1\] and the Chebyshev function \[\psi(x)=\sum_{p\leq x}\left \lfloor\frac{\log x}{\log p}\right \rfloor\log p. \] One of the early equivalences of this type, proved by Von Koch in 1901 is that the Riemann hypothesis is equivalent to \[\pi(x)= \text{Li}(x)+O(\sqrt{x}\log(x)) \text{ as } x\to\infty\] where \[ \text{Li}(x)=\int_0^{\infty}\frac{ dx}{\log x}.\] These often take the form of an inequality asserted for sufficiently large x , for example, the Riemann hypothesis is also equivalent to \[ \psi(x)=x+ O\left(x^{\frac12}\log^2 x\right) \text{ as } x\to\infty. \] The proofs of these two equivalences, as well as a great many others, are given in the book. When a proof is not included references are given where to find it. While these two equivalents have clear connection to the primes, there are others whose connection is not as immediately evident. One of my favorites is the symmetric group criterion, discovered only in the late 80s. The Riemann hypothesis is equivalent to \[ \log g(n)<\sqrt{Li^{-1}(n)} \textrm{ (for sufficiently large n) }\] where \(g(n)\) is the maximum order of an element of the symmetric group \(S_n\) and \(Li^{-1}\) is the inverse function of \(Li\). At the end of each of the chapters of the first volume are open problems.

Volume Two is devoted to analytic equivalents. As an early example, M. Riesz proved in 1916 that the Riemann Hypothesis is equivalent to the condition \[ R(x):= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{(n-1)!\zeta(2n)}x^n \ll_\epsilon x^{\frac14 +\epsilon}. \] The topics in Volume Two contains both classical material as well as more modern developments. This volume broaches some of the more diverse range of fields as mentioned above. Chapter 12 is the sole chapter on the generalized Riemann hypothesis.

There is a suite of Mathematica software available from the author for further numerical explorations. RHpack is for the classical Riemann hypothesis while GHpack is for the generalized Riemann hypothesis.

In both volumes, some of the equivalents are to a greater or lesser degree foundational, in the sense that some more modern equivalences are consequences of older established equivalences, although their proofs can be non-trivial. Attention is given to this when appropriate.

As the MAA Reviews are especially interested in possible uses for undergraduates, it should be pointed out again that these volumes are aimed at an audience that includes graduate students and researchers and that a good graduate course in analysis would probabily be a minimal prerequisite for most readers. Nonetheless a resourceful undergraduate research mentor might be able to find some things that might be accessible to advanced undergraduates. For example, the fact that the Euler product for \(\zeta(s)\) diverges when \(s=1\), entails \(\sum\frac{1}{p}=\infty\) and so the infinitude of primes (this appears as an exercise in “Baby Rudin”). Use of some of the RHpack and GHpack software is something that could possibly used in projects for undergraduates.

Being a first printing, the books do have some typos/misprints, some rather glaring. The author has a list of errata on his web page. It would have been nice to have a table of symbols but all in all these two volumes are a must have for anyone interested in the Riemann Hypothesis.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

1. Introduction
2. Series equivalents
3. Banach and Hilbert space methods
4. The Riemann Xi function
5. The de Bruijn-Newman constant
6. Orthogonal polynomials
7. Cyclotomic polynomials
8. Integral equations
9. Weil's explicit formula, inequality and conjectures
10. Discrete measures
11. Hermitian forms
12. Dirichlet L-functions
13. Smooth numbers
14. Epilogue
Appendix A. Convergence of series
Appendix B. Complex function theory
Appendix C. The Riemann-Stieltjes integral
Appendix D. The Lebesgue integral on R
Appendix E. Fourier transform
Appendix F. The Laplace transform
Appendix G. The Mellin transform
Appendix H. The gamma function
Appendix I. Riemann Zeta function
Appendix J. Banach and Hilbert spaces
Appendix K. Miscellaneous background results
Appendix L. GRHpack mini-manual