Ergodic theoretic proof of equidistribution of hecke. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to. Other readers will always be interested in your opinion of the books youve read. Equidistribution of signs for hilbert modular forms of halfintegral weight. Written for graduate students and researchers alike, this set of lectures provides a structured introduction to the concept of equidistribution in number theory. I know that duke 88 proved equidistribution for all cm points and that this has since been refined in various ways, but i couldnt find any. I amaiming tocover basicresultslikelinniksanddukestheorems,aswellascertainaspectsof. If n 1 is an integer and a is relatively prime to n, then there are in. Arithmetic, computational and probabilistic aspects 29 apr 17 may 2019. Main equidistribution in number theory, an introduction. We investigate the behaviour of the sequence log pnnen. Lfunctions and equidistribution theorems in number theory. Why the irs cares about the riemann zeta function and number. He be came interested in ergodic theory, because they could prove hard theorems, equidistribution is a powerful tool in number theory.
Past milestones in the subject include the proof of the maninmumford conjecture given by szpiroullmozhang, and bakerdemarcos work on the. With more than 2,200 courses available, ocw is delivering on the promise of open sharing of knowledge. Buy equidistribution in number theory, an introduction nato science series ii. Equidistribution in homogeneous spaces and number theory. We deal with shifted convolution sums as in hol09, with various simplifications in our analysis due to the knowledge of the ramanujanpetersson conjecture in this holomorphic case. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complexvalued. An introduction andrew granville, andrew granville, zeav rudnick. Equidistribution in homogeneous spaces and number theory elon lindenstrauss abstract. Introduction theory of benfords law applications conclusions refs. Michael mascagni applied and computational mathematics division, information technology laboratory national institute of standards and technology, gaithersburg, md 208998910 usa and department of computer science department of mathematics department of scienti. In the language of algebraic geometry, these points should be zariski dense in the a.
Book 237 kindle edition by andrew granville, zeev rudnick. We use fourieranalytic methods to give a new proof of bilus theorem on the complex equidistribution of small points on the onedimensional algebraic torus. Pgl 2 acts on the latter by linear change of variables, twisted by inverse determinant. In this paper we will examine one such proof technique which sees use in the proof of a remarkable result in number theory. Download it once and read it on your kindle device, pc, phones or tablets. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Equidistribution and benfords law equidistribution yn. Equidistribution in number theory, an introduction andrew granville, zeev rudnick eds. We survey some aspects of homogeneous dynamics the study of algebraic group actions on quotient spaces of locally compact groups by discrete subgroups. For an explanation of the background pattern, skip ahead to the end of the page.
Ergodic theoretic proof of equidistribution of hecke points volume 26 issue 1 alex eskin, hee oh skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. We also discuss a few interesting results that follow from weyls theorem. Equidistribution and weyls criterion weyls equidistribution. Ergodic theoretic proof of equidistribution of hecke points. Illustration of filling the unit interval horizontal axis with the first n terms using the equidistribution theorem with four common irrational numbers, for n from 0 to 999 vertical axis. From july 11th to july 22nd, 2005, a nato advanced study institute, as part of the series seminaire.
Plots in cof the values of generalized kloosterman sums. In the mid1970s, selberg gave a lecture at one of rockefeller universitys number theory days under the general heading of spectral theory and equidistribution properties of discrete groups. Equidistribution in number theory, an introduction springerlink. An introduction to the linnik problems springerlink.
This talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. Joint equidistribution of cm points annals of mathematics. We give special emphasis to results pertaining to the distribution of orbits of explicitly. Feb 26, 2018 exponents for the equidistribution of shears and applications article pdf available in journal of number theory february 2018 with 22 reads how we measure reads. The story of equidistribution in number theory began about a hundred years ago with h. An introduction andrew granville, andrew granville, zeav rudnick written for graduate students and researchers alike, this set of lectures provides a structured introduction to the concept of equidistribution in number theory. Kesten in ke1, ke2, as well as in his subsequent articles on similar probabilistic number theory topics, works in the setting analogous to that of ho1. By connecting dynamical systems and number theory, this graduate textbook on ergodic theory acts as an introduction to a. Pdf notes on equidistribution in number theory alex best. The orange ball marks our current location in the course.
Introduction to analytic number theory fall 2015 if you find a mistake, omission, etc. The proof makes use of the following elementary criterium for equidistribution. The same material was included in a lecture at stanford in 1980. Equidistribution techniques in arithmetic dynamics. There were about one hundred participants from sixteen countries around the world. Equidistribution and weyls criterion by brad hannigandaley we introduce the idea of a sequence of numbers being equidistributed mod 1, and we state and prove a theorem of hermann weyl which characterizes such sequences. We approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve. Equidistribution and primes presented by peter sarnak professor of mathematics institute for advanced study peter sarnak has made major contributions to to number theory, and to questions in analysis motivated by number theory. Pdf doc del clipboard journal article mr2508649katz, nicholas m.
Equidistribution in number theory, an introduction. As a consequence we establish for arbitrary products of indefinite shimura curves the equidistribution of galois orbits of generic sequences of cm points all of whose components have the same fundamental discriminant, assuming the cm fields are split at two fixed primes and have no exceptional zero. The questions that we discuss are generalizations of the twin prime conjecture. Application to prime number theory most of the deep, exciting applications of weyls theo rem require a knowledge of abstract measure theory see 1 or of number theory. Equidistribution and primes 2 1 i have chosen to talk on this topic because i believe it has a wide appeal and also there have been some interesting developments in recent years on some of these classical problems. Equidistribution and benfords law equidistribution fyng1 n1 is equidistributed modulo 1 if probability yn mod 1 2 a. Sieving for mass equidistribution annals of mathematics. Why the irs cares about the riemann zeta function and. Mar 03, 2020 this talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. An elementary proof for the equidistribution theorem the mathematical intelligencer september 2015, volume 37, issue 3, pp 12. We confirm a conjecture of marklof regarding the limiting distribution of certain sparse collections of points on expanding horospheres. Proceedings of the nato advanced study institute on equidistribution in number theory, montreal, canada, july 1122, 2005. Introduction to analytic number theory introduction to. On the expansions of a real number to two distinct bases.
Before i list some applications of equidistribution, i need to maek clear what equidistribution is and is not. Determining hilbert modular forms by central values of rankinselberg convolutions. Newest equidistribution questions mathematics stack exchange. Can we say anything about the asymptotic equidistribution of the corresponding cm points on the classical modular curve.
Equidistribution in number theory, an introduction andrew. Introduction to analytic number theory by ram murty 2015 graduate mathematics. The mission of the institute is to foster mathematical research, both fundamental and multidisciplinary, in particular, research that links mathematics to other disciplines, to nurture the growth of mathematical expertise among research scientists, to train talent for research in the mathematical sciences, and to serve as a platform for research interaction between the. Especially what it is, because many people have a rather narrow view of it. Mit opencourseware makes the materials used in the teaching of almost all of mits subjects available on the web, free of charge. Equidistribution and primes peter sarnak princeton math. In fact, we attempt to write the proofs so that they use as little arithmetic as possible, in order to clearly isolate the crucial numbertheoretic ingredients which are involved. The point of these notes is not in t he rst place to serve as. Weyl, a doyen of early twentieth century mathematics, presented in. Equidistribution implies density, but is so much more, for example we.
Lfunctions and equidistribution theorems in number theory daniel miller january 28, 2014 1 dirichlet and chebotarev recall the following classical result of dirichlet on primes in arithmetic progressions. Especially what it is, because many people have a rather narrow view of it first of all, equidistribution is neither a subarea of number theory or analysis nor a technique. Arithmetic randonn ee an introduction to probabilistic. Introduction to analytic number theory introduction to exponential sums. I know that duke 88 proved equidistribution for all cm points and that this has since been refined in various ways, but i couldnt find any literature addressing this question. The motivation for this question comes from two different sources. We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m 2 k p 1 p 2 p s, where k and s are nonnegative integers and p j are distinct fermat primes. This paper is a slightly enlarged version of a series of lectures on the linnik problems given at the smsnato asi 2005 summer school on equidistribution in number theory. There were about one hundred participants from sixteen. Equidistribution of polynomial sequences in function fields, with applications thai ho ang l e and yuru liu abstract. First of all, equidistribution is neither a subarea of number theory or analysis nor a technique. Equidistribution of primitive rational points on expanding. The central problem of the equidistribution theory of holomorphic curves, crudely stated, is the following. Equidistribution in number theory, an introduction nato science series ii.
Quantitative equidistribution results are also discussed. The purpose of this proposal is to investigate from various perspectives some equidistribution problems associated with homogeneous spaces of arithmetic type. His interest in mathematics is wideranging, and his research focuses on the theory of zeta functions and. There are far more good problems than we had time for in our schedule. It is for this reason we decided to hold a school on equidistribution in number theory to introduce junior researchers to these beautiful questions, and to determine whether di. Why the irs cares about the riemann zeta function and number theory and why you should too. Fourier theory and its extensions, a great number of powerful techniques were produced incidentally. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. Equidistribution, lfunctions and ergodic theory 3 which identi. We prove a function eld analog of weyls classical theorem on equidistribution of polynomial sequences. The orange ball marks our current location in the course for an explanation of the background pattern, skip ahead to the end of the page. New equidistribution estimates of zhang type article pdf available in algebra and number theory 89. The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. Analytic number theory lecture notes based on davenports book andreas str ombergsson these lecture notes follow to a large extent davenports book 15, b ut with things reordered and often expanded.
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