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In Search of the Riemann Zeros

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The Riemann hypothesis on the location of the zeros of the Riemann zeta function is still an open problem despite almost 150 years of intense effort by some of the world's finest mathematicians. The problem does not require a lot of mathematical background to understand, but some of the mathematical tools that have been developed for its resolution are very deep and require years of study for proper understanding.

This book, which is full of highly interesting ideas but is for the main part very speculative, outlines an approach to the Riemann hypothesis that connects very disparate areas of both mathematics and physics. These areas include string theory, noncommutative geometry, number theory, operator algebras, and quantum statistical physics. To appreciate even a small portion of any one of these areas would be time-consuming, and so individuals who decide to study this book must have an atypical background and large blocks of time. Due to its speculative nature, it is different than most other mathematical books that emphasize rigor and formal expression, but its virtue is that it contains a wealth of ideas that could be of interest to those readers who are searching for non-trivial research problems.

Loosely speaking, the author wants to resolve the Riemann hypothesis by connecting arithmetic to geometry, or more properly, with "noncommutative" geometry. To give this approach some plausibility, he begins the book with an outline of some concepts from `string theory on a circle' and how the accompanying notion of T-duality has an analogy with the Riemann zeta function. The main point behind this discussion, and the most interesting one, is that the familiar functional equation for the Riemann zeta function is an analog of target space duality and that the Riemann hypothesis is related to the notion of a fundamental (or minimum) length in string theory. The Riemann hypothesis is thus connected intimately with considerations from physics.

After this heuristic introduction, the author spends the rest of the book detailing his approach to the Riemann hypothesis, which depends essentially on the following concepts:
* The `fractal string': This is a nonempty open set of the real line consisting of the disjoint union of open intervals (Om). Each connected component of this set has a finite length, and the lengths are ordered by multiplicity. Particular attention is directed towards the boundary of the fractal string and the volume of the collection of points, called the `tubular neighborhood' of the fractal string, which are a small distance away from the boundary. Also of importance is the `Minkowski dimension' of the fractal string (some readers will know this as the `capacity' or `box-counting' dimension).
* The `geometric zeta function' of the fractal string is then defined as an infinite Dirichlet series of its lengths, with the restriction that the real parts of the exponents (s) in the terms of the series are strictly larger than the Minkowski dimension (Dm). This restriction is placed in order that the zeta function is holomorphic. When s = Dm, one can use analytic continuation to obtain a meromorphic geometric zeta function. The Minkowski dimension is chosen, rather than other notions of (fractal) dimension such as the Hausdorff dimension, in order that the Dm is invariant under arbitrary rearrangements of the connected components.
* The `spectrum' of the fractal string is defined as the spectrum of the Laplacian acting on the space of square integrable functions of Om. This spectrum is discrete and infinite.
* The `spectral zeta function', as a function of s, is the sum over all elements of the spectrum with exponent -s, and can be written as the product of the geometric zeta function and the (classical) Riemann zeta function (The Riemann zeta function appears here since it is the spectral zeta function of the unit interval).
* The `fractal membrane': If L is sequence of positive real numbers Lj strictly less than one then a fractal membrane is an infinite product (the author calls this the "adelic" product because he wants to connect it with number theory) of open intervals each with length 1/Log(1/Lj). A Hilbert space of square-integrable wave functions is assigned to each open interval. The Hilbert space associated with the fractal membrane is then the (restricted) tensor product of all these Hilbert spaces. A `vacuum vector' of this Hilbert space is defined as the zero mode of an operator acting on this Hilbert space, this operator being the "square root" of the Laplacian with Neumann boundary conditions on each Lj.
* Interpret the Euler product representation of the Riemann zeta function as the partition function of a fractal membrane.
* View the fractal membrane as an adelic Riemann surface with infinite genus and attempt to associate to it a noncommutative stringy space (NCSS). The author does not do this explicitly but instead gives a few heuristic arguments as to how it could be accomplished.
* Construct the moduli space of fractal membranes and view it as a noncommutative space in terms of the Connes theory of noncommuative geometry. The moduli space M(fm) of fractal membranes is defined in terms of a quotient space of the collection of all nonincreasing sequences of positive real numbers less than 1 by an equivalence relation on the spectra of fractal membranes.
* Use M(fm) to classify the sets of poles and zeros of the partition functions of fractal membranes.
* Interpret the concept of an "arithmetic site" as the "core" (in the author's terminology) of the moduli space of fractal membranes. This moduli space is acted upon by a 1-parameter group of automorphisms called the "modular" or "Frobenius" flow.
* Use the modular theory of von Neumann algebras (W*-algebras) to establish a conjecture giving a relationship between modular flows and the Riemann hypothesis. The statement of this conjecture and the summary of the classification theory of factors sheds more light on what the author means by the "core" of the moduli space of fractal membranes. The Takesaki theory of von Neumann algebras proves that a factor of type III(1) can be written as the crossed product of a type II(infinity) factor (called the core) and a 1-parameter group of automorphisms called the multiplicative flow. The author conjectures that the moduli space of membranes is a factor of type III(1) and thus can be written as such a crossed product. The core of this crossed product is conjectured to be a "noncommutative" curve and an "arithmetic site". The multiplicative flow is conjectured to be a "Frobenius flow" such that the flow of zeta functions induced by this flow "condenses" on the core so that their zeros correspond to the critical line.

The concepts needed to discuss this strategy are discussed in varying degrees of detail in the book, as well as the differences between `fractal' membranes and `self-similar' ones. The main theme in the book is how to study fractal membranes using noncommutative geometry, especially the constructions of the mathematician Alain Connes. Readers who want to appreciate the author's discussions will be greatly assisted by reading the book by Alain Connes on noncommuative geometry. In that book Connes shows how to study foliations using a noncommuative version of measure theory, and he does so in a way that is very understandable. Connes' book is a good prelude to this one, but readers are still expected to have a thorough knowledge of the theory of von Neumann algebras, particularly the classification theory of factors and its generalization by Alain Connes. In particular, readers are to understand why noncommutative von Neumann algebras are "dynamical objects" to the extent that they always possess a 1-parameter group of automorphisms, a property that is not true in the commutative case.