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2 July, 2008 at 6:28 pm Terence Tao It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.

3 July, 2008 at 3:41 am Gergely Harcos I also have some (perhaps milder) troubles with the proof. It seems to me as if Li had treated the Dirac delta on L^2(A) as a function. For example, the first 5 lines of page 28 make little sense to me. Am I missing something here?

4 July, 2008 at 5:15 am Lior Silberman The function defined on page 20 does have a strong dilation symmetry: it is invariant by multiplication by ideles of norm one (since it is merely a function of the norm of ). In particular, it is invariant under multiplication by elements of . I’m probably missing something here. Probably the subtlety is in passing from integration over the nice space of idele classes to the singular space . The topologies on the spaces of adeles and ideles are quite different. There is a formal error in Theorem 3.1 which doesn’t affect the paper: the distribution discussed is not unique. A distribution supported at a point is a sum of derivatives of the delta distribution. Clearly there exist many such with a given special value of the Fourier transform. There is also something odd about this paper: nowhere is it pointed out what is the new contribution of the paper. Specifically, what is the new insight about number theory?

References: Cipra, B. 1998. A prime case of chaos. In What's Happening in the Mathematical Sciences, Vol. 4. Providence, R.I.: American Mathematical Society. (Available at http://www.ams.org/new-in-math/happening.html.) ______. 1996. Prime formula weds number theory and quantum physics. Science 274(Dec. 20):2014. Davis, P.J., and R. Hersch. 1981. The Mathematical Experience. New York: Viking Penguin. Katz, N.M., and P. Sarnak. 1999. Zeroes of zeta functions and symmetry. Bulletin of the American Mathematical Society 36(January):1. Peterson, I. 1995. Cavities of chaos. Science News 147(April 29):264. Richards, I. 1978. Number theory. In Mathematics Today: Twelve Informal Essays. L.A. Steen, ed. New York: Springer-Verlag. Peter Sarnak's lecture on random matrix models in number theory and quantum mechanics is available at http://www.msri.org/publications/video/fall98/mandm.html. Andrew Odlyzko's Web page at http://www.research.att.com/~amo/ features computations of the zeros of the zeta function.

The Riemann hypothesis was first proposed in 1859 by the German mathematician Georg Friedrich Bernhard Riemann (1826-1866). It concerns the so-called zeta function, which encodes a great deal of information about the seemingly haphazard distribution of prime numbers among the integers (see The Mark of Zeta, June 19, 1999).

M. Anshel and D. Goldfeld, "Zeta Functions, One-Way Functions, and Pseudorandom Number Generators", Duke Mathematical Journal, Vol. 88 No. 2 (1997) 371-390. "In 1997,Anshel and Goldfeld [6],presented an explicit construction of a pseudorandom number generator arising from an elliptic curve,which can be effectively computed at low computational cost. They introduced a new intractable problem,distinct from integer factorization or the discrete log problem, that leads to a new class of one-way functions based on the theory of zeta functions,and against which there is no known attack."- Richard M. Mollin,"Introduction to Cryptography" CRC Press (2000)

We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at . We give two separate proofs of this result, with the more general method not restricted to . We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence.

A new mathematical object, an elusive cousin of the Riemann zeta-function, was revealed to great acclaim recently at the American Institute of Mathematics. Ce Bian and Andrew Booker from the University of Bristol showed the first example of a third degree transcendental L-function.

Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:   where q is an integer called the level, d is the degree, and the numbers  are Langland's parameters. is an analytic continuation of the factorial function  that is valid not only for integers but all complex numbers. There are two types of L-functions: algebraic and transcendental. These are classified according to their degree. If the Langland's parameters are rational or algebraic (that is, are complex numbers that are roots of non-zero polynomials with rational coefficients), then the L-function is algebraic. If these numbers are transcendental (that is, non-algebraic, such as  or  , then the L-function is transcendental. The Riemann zeta-function is the L-function where the level is 1, the degree is 1 and the Langland's parameters are 0 — that is, a first degree algebraic L-function. The Bristol researchers showed the first example of a third degree transcendental L-function.

The primary aim of this book is to create situations in which the zeta function, or other L-functions, will appear in spectral-theoretic questions. A secondary aim is to connect pseudo-differential analysis, or quantization theory, to analytic number theory. Both are attained through the analysis of operators on functions on the line by means of their diagonal matrix elements against families of arithmetic coherent states: these are families of discretely supported measures on the line, transforming in specific ways under the part of the metaplectic representation or, more generally, representations from the discrete series of SL(2,R), lying above an arithmetic group such as SL(2,Z).

The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to state a number of conjectures.

In recent years the application of random matrix techniques to analytic number theory has been responsible for major advances in this area of mathematics. As a consequence it has created a new and rapidly developing area of research. The aim of this book is to provide the necessary grounding both in relevant aspects of number theory and techniques of random matrix theory, as well as to inform the reader of what progress has been made when these two apparently disparate subjects meet. This volume of proceedings is addressed to graduate students and other researchers in both pure mathematics and theoretical physics. The contributing authors, who are among the world leading experts in this area, have taken care to write self-contained lectures on subjects chosen to produce a coherent volume.• Self-contained lectures by world-leading experts in the field • The volume is integrated, indexed and cross-referenced • This title covers the most important and recent advances in the subjectContents1. Introduction; 2. Prime number theory and the Riemann zeta-function; 3. Notes on pair correlation of zeros and prime numbers; 4. Notes on eigenvalue distributions for the classical compact groups; 5. Compound nucleus resonances, random matrices and quantum chaos; 6. Families of L-functions and 1-level densities; 7. Basic analytic number theory; 8. Applications of mean value theorems to the theory of the Riemann zeta function; 9. L-functions and the characteristic polynomials of random matrices; 10. Mock gaussian behaviour; 11. Some specimens of L-functions; 12. Computational methods and experiments in analytic number theory.

The connections between the zeros of the zeta-function and random matrix theory have become the most active and exciting threads of research in the hunt for the Riemann hypothesis. Rockmore devotes four chapters at the end of his book to various aspects of this research. He discusses the work of Sarnak and Katz on analogous results for function fields. He also discusses work of Tracy, Widom, and Deift that connects the distribution of eigenvalues of random matrices to properties of permutations. This chapter has the engaging title "God May Not Play Dice, but What About Cards?"

The analytic theory of algebraic numbers H. M. Stark Source: Bull. Amer. Math. Soc. Volume 81, Number 6 (1975), 961-972. Primary Subjects: 12–02, 12A50, 12A70 Full-text: Access granted (open access) PDF File (1009 KB)

Among the modern methods used to study prime numbers, the ‘sieve’ has been one of the most efficient. Originally conceived by Linnik in 1941, the ‘large sieve’ has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.• Explores new and surprising applications of the large sieve method, an important technique of analytic number theory • Presents applications in fields as wide ranging as topology, probability, arithmetic geometry and discrete group theory • Motivated, clear and self-contained discussions introduce readers to a technique previously confined to one fieldContentsPreface; Prerequisites and notation; 1. Introduction; 2. The principle of the large sieve; 3. Group and conjugacy sieves; 4. Elementary and classical examples; 5. Degrees of representations of finite groups; 6. Probabilistic sieves; 7. Sieving in discrete groups; 8. Sieving for Frobenius over finite fields; Appendix A. Small sieves; Appendix B. Local density computations over finite fields; Appendix C. Representation theory; Appendix D. Property (T) and Property (τ); Appendix E. Linear algebraic groups; Appendix F. Probability theory and random walks; Appendix G. Sums of multiplicative functions; Appendix H. Topology; Bibliography; Index.

Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of ‘partitio numerorum’. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1–70 (sec. 3)] on the asymptotic behavior of π2r(x), the number of prime pairs with p≤x. Assuming Riemann’s Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta’s complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta’s complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery’s work on pair-correlation, would imply that there is an abundance of prime pairs.