Expected properties of the flow s of zeros and zeta functions 243. The riemann zeta function and conness noncommutative geometry 29th canadian symposium on operator algebras april 26, 2001 may 01, 2001 april 30, 2001 01. Riemannian, quantum and noncommutative geometry 2 1. In this book, the author proposes a new approach to. A dynamical interpretation of the riemann hypothesis 241 5. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. The paper used in this book is acidfree and falls within the guidelines. This reduces the riemann hypothesis to the validity of the trace formula and eliminates. 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 worlds finest mathematicians. Dimensional regularization and noncommutative geometry. The riemann zeta function is defined as the analytic continuation of the function defined for.
The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role. Connes, trace formula in noncommutative geometry and the zeros of the riemann zeta function, selecta mathematica, new series 5, n. Trace formula in noncommutative geometry and the zeros of the riemann zeta function alain connes abstract. In essence this work restates the riemann hypothesis in terms of mathematical language more commonly associated to physics, and is part of connes pioneering work in noncommutative geometry, a new area of mathematics also closely related to physics, in particular quantum mechanics and quantum field theory. While it was not created by riemann, it is named after him. We give a spectral interpretation of the critical zeros of the riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. On the meanvalue and omegatheorems for the riemann zeta function. On the meanvalue and omegatheorems for the riemann zetafunction.
His reformulation builds upon earlier joint work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Jun 25, 2007 moreover, with a bit more of technical effort cfr. Eudml trace formula in noncommutative geometry and the. I shall then explain that this leads to a natural spectral interpretation of the zeros of the riemann zeta function and a geometric framework in which the frobenius, its eigenvalues and the lefschetz formula interpretation of the explicit formulas continue to hold even for number fields. Noncommutative geometry and the riemann zeta function. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a riemannhilbert correspondence. Next, theres truly spellbinding material on riemanns zeta function and noncommutative geometry chapter 2, at only about 100 pages, followed by quantum statistical mechanics connected to a good deal of arithmetic and algebraic geometry everything from class field theory and kroneckerweber to elliptic curves and shimura varieties and. Michel l lapidus formulated in 1859, the riemann hypothesis is the most celebrated and multifaceted open problem in mathematics. Leonhard euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the basel problem. We give a geometric interpretation of the explicit formulas of number. What is new with connes approach to the riemann hypothesis. Spectral theory of the riemann zeta function, by yoichi motohashi, cambridge university press, 1997.
Frontiers in number theory physics and geometry ii. Pdf trace formula in noncommutative geometry and the zeros. Alain connes expository paper on the subject matter of the course, describing his motivations and some background. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Show full abstract noncommutative geometry can be used as the starting point for a geometric and physical model for the riemann zeta function.
Trace formula in noncommutative geometry and the zeros of the. The second part of the book gives an interpretation of the weil explicit formula as a trace formula and a spectral realization of the zeros of the riemann zeta function. Msri workshop schedules the riemann zeta function and. This english version of the pathbreaking french book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by alain connes. The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. Marcolli noncommutative geometry, dynamics and infinitelyadic arakelov geometry selecta math. The problem does not require a lot of mathematical background to understand, but some of the mathematical tools that have been developed for its. An introduction to the theory of the riemann zeta function, by s. We want to study the space of leaves vfinitially only as a measure space, but then successively in the more re.
Riemann formulated his famous hypothesis in 1859 in a foundational paper 31, just 8 pages in length, on the number of primes less than a given magnitude. In 64 we gave a spectral realization of the zeros of the riemann zeta function and of lfunctions based on the above action of the idele class group on the noncommutative space of adele classes. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. Spectral theory of the riemann zetafunction, by yoichi motohashi, cambridge university press, 1997. Noncommutative geometry, quantum fields and motives alain. Noncommutative geometry, quantum fields, and motives alain. It is available for downloading in any of the following formats. Topics include riemann s main formula, the prime number theorem, the riemann. What is the best source to study riemann zeta function. The noncommutative space of penrose tilings and quasicrystals. The riemann zeta function and noncommutative geometry.
In question details i see broad topics, i ll pin point sub topicsexercise that directly correspond to rzf. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative. Access full article top access to full text full pdf how to cite top. This book is the first of two volumes resulting from that meeting. The grand riemann hypothesis generalises not only the familiar rh but also the generalised rh, as it concerns all automorphic l functions, which include all. Connes, noncommutative geometry and the riemann zeta function from mathematics. Quantum statistical mechanics over function fields noncommutative tori with real multiplication the riemann zeta function and quantum mechanics adeles, ideles and gauge theory endomotives the dual system and quantum statistical mechanics frobenius and scaling the spectral realization the weil explicit formula as a trace formula.
An introduction to the theory of the riemann zetafunction, by s. Abstract we give a spectral interpretation of the critical. 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. Journees equations aux derivees partielles 1997 page 128. Buy noncommutative geometry, quantum fields and motives colloquium. The riemann hypothesis for function fields by machiel van. Connes trace formula in noncommutative geometry and the zeros of the riemann zeta function, selecta mathematica new ser. The generalised riemann hypothesis concerns all dirichlet lfunctions, of which the riemann zeta function is a single example, similarly requiring their zeros to lie on the critical line. In this book, the author proposes a new approach to understand and possibly solve the riemann hypothesis. The riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series. Marcolli, lectures on arithmetic noncommutative geometry. This reduces the riemann hypothesis to the validity of the trace formula and. Noncommutative geometry, quantum fields and motives book. Trace formula in noncommutative geometry and the zeros of the riemann zeta function alain connes abstract we give a spectral interpretation of the critical zeros of the riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances.
Interactions between number theory and operator algebras in the. Importance of the zeta function the zeta function is a very important function in mathematics. The reader will encounter many important aspects of the theory, such as bombieris proof of the riemann hypothesis for function fields, along with an explanation of the connections with nevanlinna theory and noncommutative geometry. Edwards 16, which also includes a translation of riemanns. It also presents a model of elementary particle physics based on noncommutative geometry. The riemann zeta function and noncommutative geometry 1. Edwards book riemanns zeta function 1 explains the historical context of riemanns paper, riemanns methods and results, and the. Trace formula in noncommutative geometry and the zeros of the riemann zeta function. Leonhard euler considered the above series in 1740 for positive integer values of s, and later chebyshev extended the definition to re s 1. In essence, it states that the primes are distributed as harmoniously as possibleor. Connes, trace formula in noncommutative geometry and the zeros of the riemann zeta function, selecta math. Renormalization in quantum field theory and the riemann hilbert problem ii, the beta function, diffeomorphisms and the renormalization group 286 kb 9. Noncommutative geometry and number theory tamu math.
We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of adele classes. Noncommutative geometry, quantum fields and motives in nlab. Noncommutative geometry, quantum fields and motives. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of qlattices, and is dual to a. I shall then explain that this leads to a natural spectral interpretation of the zeros of the riemann zeta function and a geometric framework in which the frobenius. The above series is a prototypical dirichlet series that. Pdf noncommutative geometry download ebook for free.
Sarnak, random matrices, frobenius eigenvalues and. In search of the riemann zeros strings, fractal membranes. The poisson summation formula, with applications 9. Trace formula in noncommutative geometry and the zeros. Noncommutative geometry, quantum fields, and motives. Noncommutative geometry and the riemann zeta function alain connes according to my rst teacher gustave choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by ones failure than anything else.
The riemann hypothesis and the geometry of the primes 6 1. The riemann zeta function and noncommutative geometry341. Trace formula in noncommutative geometry and the zeros of. Motivations, objectives and organization of this book 9 chapter 2. I shall then explain that this leads to a natural spectral interpretation of the zeros of the riemann zeta function and a geometric framework in which. We give a geometric interpretation of the explicit. Noncommutative geometry, trace formulas and the zeros of.