Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4222-5/hbk). xxvii, 558 p. $ 79.00 (2008).

This is a very nice book, interesting in many respects. The author declares it to be the result of more than fifteen years of reflection and research on or around the subject of the Riemann zeros, the celebrated Riemann Hypothesis (or Riemann’s Conjecture). The importance of this `million dollar problem’ is paramount, not just for historical reasons or for what strictly states, but for the so many desirable and important consequences, in mathematics and beyond, associated with its proof. Actually, as recognized by the author, few experts now doubt that it is true, so that in a way, to prove that it is false would be even more interesting. This is not likely to happen, since it has been numerically verified up to very large values of the argument, no less than two trillion. The book is not a traditional mathematical research monograph. The author does not claim to provide a complete solution to the original issue of its title, let alone full proofs or even partial justifications for the main proposals and conjectures that appear in the text. At best, in many cases, the author can only offer heuristic arguments based on mathematical and/or physical analogies. It should be understood from the context in the book itself (either in the text itself or in the notes) whether a given claim is a physical or heuristic statement, a reasonable expectation, a conjecture, a mathematical theorem, or neither. For example, at the present stage of knowledge, the existence of the modular flow, which is one of the authors main original issues in the book, and its expected properties are purely conjectural. The author is only sincere on this and I allow myself to transcribe all this description here so that the reader will not be later disappointed. One of the book’s strong values is its interdisciplinarity, in special in that it relates different domains of mathematics with different domains of theoretical physics. In his search of the Riemann zeros, the author works on analogies with physical theories and constructs (as string theories and dualities, as reformulated in the language of vertex algebras and noncommutative geometry, conformal field theories, quantum statistical physics, renormalization group flow) and on mathematical concepts and theories (as moduli spaces of quantized fractal strings, the author and his collaborators’ theory of complex fractal dimensions, Deninger’s spectral interpretation program and heuristic notion of `arithmetic site’, modular theory in operator algebras, and Connes’ noncommutative geometry). In some sense, also as suggested by the author, the book should be viewed partly as a research program to pursue (rather than to complete) the quest announced in its title, and partly as a contribution to a continuing dialogue between mathematicians, physicists and other geometers of the world we live in. As such, it is written in multiple tongues, sometimes in mathematical language and sometimes in physical language. Appeals to both rigor and intuition alternate in the book, in no particular order, without apparent rhyme or reason. By the way this can be fascinating for a class of readers, in search of information coming from very different sides, but not so much for some other readers who would prefer a more rigorous, step after step advance towards the desired goal. The difficulty of the second approach may provide a partial justification for choosing the first, even to this second class of readers. Just as importantly, even within the more mathematical discussions in the book, the boundaries between the traditional research areas are often blurred (this is also what I call true interdisciplinarity). This is one reason, the author says, to have found it necessary to include a significant amount of background material, as evidenced by the large number of appendices in the second part of the book. The reader may of course benefit from reading that material. In any case, I absolutely coincide with the author that, in order to profit most from it, the book must be approached with an open and flexible mind. The book is divided into five Chapters and 6 Appendices, in all it has some 600 pages. The contents of the same are as follows. In Chapter 1, a broad introduction is provided to several of the main themes encountered later: arithmetic geometry, noncommutative geometry, quantum physics and string theory, prime number theory and the Riemann zeta function, along with fractal and spectral geometry. In Chapter 2, it is explained how string theory on a circle or on a finite-dimensional torus can be used as the starting point for a geometric and physical model of the Riemann zeta function and other arithmetic L-series. In particular, by analogy with the key role played by the Poisson summation formula in both the physical and the arithmetic theory, the author contends that the classic functional equation satisfied by $\zeta$ corresponds to $T$-duality in string theory. This duality, a key symmetry that is not present in ordinary quantum mechanics, allows one to identify, physically and mathematically, two circular spacetimes with reciprocal radii. Furthermore, the author suggests that the Riemann Hypothesis may be related to the existence of a fundamental length in string theory. This is, in my view, one of the main conjectures in the book. In the first part of Chapter 3, some aspects of the author’s theory of fractal strings (one-dimensional drums with fractal boundary) and of the associated theory of complex dimensions are briefly reviewed. Then, the new concept of a fractal membrane, a suitable multiplicative (or quantum) analogue of a fractal string, is introduced. Heuristically, a fractal membrane can be thought of as a noncommutative Riemann surface with infinite genus or as an adelic infinite dimensional torus. It is then shown that the spectral partition function of a fractal membrane is naturally given by an Euler product, which reduces to the usual one for $\zeta$ in the case of the `prime membrane’ associated with the Riemann zeta function (or, equivalently, with the field of rational numbers). In this case, a new mathematical model is obtained (different from that of Bost and Connes) for the notion of a `Riemann gas’ introduced by the physicist B. Julia in the context of quantum statistical physics. The new but closely related concept of self-similar membrane, which corresponds to a different choice of statistics than for a fractal membrane when quantizing a fractal string, is also introduced. A dynamical interpretation of the partition functions of fractal membranes and of self-similar membranes is given. In the former case, it is discussed that the associated suspended flows may be called Riemann-Beurling flows. Indeed, the logarithms of the underlying (generalized) primes coincide with the `weights’ (or `lengths’) of the corresponding primitive orbits. The `Riemann flow’ is seen to be associated with the `prime fractal membrane’ (or, equivalently, with the field of rational numbers). In Chapter 4, various noncommutative and increasingly rich models of fractal membranes are discussed, in particular, some very recent work of the author and R. Nest in which it is shown that fractal (and self-similar) membranes are the second (or Dirac) quantization of fractal strings. In this context, the choice of Fermi-Dirac or Bose-Einstein, in a second and improved construction -- quantum (resp., Gibbs) statistics corresponds to fractal (resp., self-similar) membranes. In short, it follows that fractal membranes (or their self-similar counterparts) can truly be considered as `quantum fractal strings’. It is argued that one of the new heuristic and mathematical insights provided by the latter work is that once fractal strings have been quantized, their endpoints are no longer fixed on the real axis but are allowed to move freely within suitable copies of the holomorphic disc in the complex plane. This seems to be somewhat analogous to the role played by D-branes in string or M-theory. As explained earlier in Chap. 3, one can associate a prime fractal membrane to each type of arithmetic geometry, including algebraic number fields and function fields (for example, curves or higher-dimensional varieties over finite fields). Near the end of Chap. 4, it is proposed that a more geometric, algebraic and physical model of arithmetic geometries could be based on the `noncommutative stringy spacetime’ corresponding to closed strings propagating in a fractal membrane viewed, for example, as an adelic infinite dimensional torus. Such a spacetime could be thought of as a sheaf of ordinary noncommutative or quantum spaces and thus, in the author’s framework, of vertex operator algebras along with dual (or `chiral’) pairs of Dirac operators. The functional equation satisfied by an arithmetic zeta function such as $\zeta$ would then be the analytic counterpart of Poincaré duality at the cohomological level, and of T-duality, at the physical level. Accordingly, it is conjectured that a suitable spectral and cohomological interpretation of the (dynamical) complex dimensions of fractal membranes and, in particular, of the `Riemann zeros’, the nontrivial zeros of $\zeta$ could be obtained in this context, by means of the associated sheaf of vertex algebras. In Chap. 5, the author’s moduli spaces of fractal strings and of fractal membranes viewed as highly noncommutative spaces significantly generalizing the set of all Penrose tilings are suggested to be a natural receptacle for zeta functions and for a suitable extension of Deninger’s heuristic notion of `arithmetic site’. The author concludes by proposing a new geometric and dynamical interpretation of the Riemann Hypothesis, expressed in terms of a suitable noncommutative flow of zeta functions acting on the moduli space of fractal membranes, along with the associated flow of zeros (with each of these flows referred to as a `modular flow’ or as an `extended Frobenius flow’). The truth of the Riemann Hypothesis (and of its natural extensions) would ultimately follow from the convergence of the zeros of the corresponding zeta functions to the critical line or, equivalently, to the equator of the Riemann sphere, both from within the lower and upper hemispheres, using $T$-duality and the associated `generalized functional equations’. In the last chapter of the book, Chap. 5, analogies are drawn between the conjectured modular flows of zeta functions and of their associated zeros and other flows which arise naturally in mathematics and physics. These flows include Wilson’s renormalization flow, the Ricci flow on three-dimensional manifolds, as well as the `KP-flow’ (viewed as a noncommutative, geodesic flow). Accordingly, the modular flow of zeta functions could be viewed as a noncommutative and arithmetic analogue of the Ricci flow. Similarly, the associated flow of zeros could be thought of as an arithmetic, noncommutative KP-flow. A model of the proposed modular flows is discussed, which is called the `KMS-flow’ (for generalized Pólya-Hilbert operators) and is motivated in part by analogies with quantum statistical physics (in the operator algebraic formalism), along with the Feynman integral and renormalization flow (or group) approaches to quantum systems with highly singular interactions. The six appendices in the book are devoted to the following subjects. In Appendix A vertex algebras are discussed, their definition, translation and scaling operators, and their basic properties. Appendix B deals with the Weil conjectures and the Riemann Hypothesis, introducing with this purpose the varieties over finite fields and their zeta function and the zeta function of curves over finite fields. Appendix C is devoted to the very important Poisson summation formula, in its more general version for dual lattices, and its applications. In this respect, the modularity of theta functions, cusps forms, Hecke operators and Hecke forms, the Hecke L-series of modular forms, and modular forms of higher level and their L-functions, are discussed. Appendix D deals with generalized primes and Beurling zeta functions, and corresponding analogues of the prime number theorem and of a generalized functional equation, to finish with partial orderings on generalized integers. Appendix E is devoted to the Selberg class of zeta functions, the Selberg conjectures, and Langlands’ reciprocity laws. The last one, Appendix F, is of a more physical nature; there, in about forty pages, the concepts of noncommutative space of Penrose tilings and quasicrystals is discussed in some detail. It may be useful for the reader to be aware from the outset of the following distinction between the various parts of the book. Let me here paraphrase the author once more, since I fully agree when he says that, while Chapter 1 is intended for a `general’ scientific audience, Chapter 2 is more physics-oriented but still accessible to mathematicians not familiar with string theory, whereas the rest of the book (Chapters 3--5) is clearly of a much more mathematical (but not easy) nature, even though in various places it draws on the physical language, intuition and formalism discussed in Chapter 2. Relevant background material is provided in several places within the text, as well as in the six appendices, in order to make the book more easily accessible and facilitate the transition between its various parts. It seems clear that the author has tried to write the book in such a way that someone not familiar with all the subjects can still understand the main ideas and concepts involved, in what he has been quite successful, in my opinion. The reader should be cautioned, however, that the mathematics underlying parts of the theory presented in this book is rather formidable and, in fact, is often not yet fully developed or even precisely formulated as of now. The book is ambitious, original, and easy to read; if you have some knowledge about the subjects it deals with, you can appreciate it in all its value. It is not a textbook, neither a research monograph, it is somewhat sketchy, in attacking so many subjects -- as recognized by the author himself -- but it certainly has a value in its uniqueness as a source of a lot of information that is very much disperse in the specialized literature. It even has its poetic vein. For this I want to recall an excerpt from the preface, where the author discusses the fact that the Riemann Hypothesis can be poetically (but rather accurately) reformulated as stating that $\Bbb Q$, the field of rational numbers, lies as harmoniously as possible within the field of real numbers, $\Bbb R$. He continues to observe that since the ring of integers, $\Bbb Z$ and hence, its field of fractions, $\Bbb Q$ is arguably the most basic and fundamental object of all of mathematics (because it is the natural receptacle for elementary arithmetic), one may easily understand the centrality of the Riemann Hypothesis in mathematics and surmise its possible relevance to other scientific disciplines, especially physics. Talking about the possible dialogue at the interface between mathematics and physics, he then notes that for some physicists, only $\Bbb Q$ truly exists. Yet, in practice as well as in theory, all measurable quantities are given by real numbers, not just by rational numbers. This is in fact a subtle, very simple, but also an extremely important point that reveals two very different attitudes in trying to understand (model, describe) the world we live in. Indeed for a true physicist this last sentence has no sense: the result of an experiment will never be a real number, neither is now, nor will it be in 100, nor in $10^{500}$ years to come. Be careful, this is a seemingly trivial but very profound statement: either one understands the issue behind it, or one does not.