Lfunctions and equidistribution theorems in number theory. 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. Equidistribution in homogeneous spaces and number theory. Weyl equidistribution the \exponential in question is the complex exponential, which we normalize. Other readers will always be interested in your opinion of the books youve read. We survey some aspects of homogeneous dynamics the study of algebraic group actions on quotient spaces of locally compact groups by discrete subgroups. There are far more good problems than we had time for in our schedule. Pdf notes on equidistribution in number theory alex best. In this paper we will examine one such proof technique which sees use in the proof of a remarkable result in number theory. We also discuss a few interesting results that follow from weyls theorem.
Equidistribution in number theory, an introduction andrew granville, zeev rudnick eds. Determining hilbert modular forms by central values of rankinselberg convolutions. Equidistribution of signs for hilbert modular forms of halfintegral weight. The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complexvalued. Equidistribution and weyls criterion weyls equidistribution. 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.
He be came interested in ergodic theory, because they could prove hard theorems, equidistribution is a powerful tool in number theory. Introduction to analytic number theory by ram murty 2015 graduate mathematics. If n 1 is an integer and a is relatively prime to n, then there are in. We also provide a general and exact lower bound for the proportion of. 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. 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. Equidistribution implies density, but is so much more, for example we.
By connecting dynamical systems and number theory, this graduate textbook on ergodic theory acts as an introduction to a. Joint equidistribution of cm points annals of mathematics. Equidistribution in homogeneous spaces and number theory elon lindenstrauss abstract. Fourier theory and its extensions, a great number of powerful techniques were produced incidentally. Arithmetic, computational and probabilistic aspects 29 apr 17 may 2019. We give special emphasis to results pertaining to the distribution of orbits of explicitly. 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 primes peter sarnak princeton math. Ergodic theoretic proof of equidistribution of hecke points. We investigate the behaviour of the sequence log pnnen.
Especially what it is, because many people have a rather narrow view of it. The purpose of this proposal is to investigate from various perspectives some equidistribution problems associated with homogeneous spaces of arithmetic type. We prove a function eld analog of weyls classical theorem on equidistribution of polynomial sequences. Quantitative equidistribution results are also discussed. Buy equidistribution in number theory, an introduction nato science series ii. Equidistribution and benfords law equidistribution fyng1 n1 is equidistributed modulo 1 if probability yn mod 1 2 a. The orange ball marks our current location in the course. Newest equidistribution questions mathematics stack exchange. 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. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to. Before i list some applications of equidistribution, i need to maek clear what equidistribution is and is not. Why the irs cares about the riemann zeta function and number. Uniform distribution granville 1 lecture, rudnick 1 lecture. Sieving for mass equidistribution annals of mathematics.
Can we say anything about the asymptotic equidistribution of the corresponding cm points on the classical modular curve. 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. An introduction to the linnik problems springerlink. 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. 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. The same material was included in a lecture at stanford in 1980. The story of equidistribution in number theory began about a hundred years ago with h. 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. Equidistribution techniques in arithmetic dynamics. In the language of algebraic geometry, these points should be zariski dense in the a. We approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve. Mit opencourseware makes the materials used in the teaching of almost all of mits subjects available on the web, free of charge. The central problem of the equidistribution theory of holomorphic curves, crudely stated, is the following. 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.
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. 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. Equidistribution of polynomial sequences in function fields, with applications thai ho ang l e and yuru liu abstract. Introduction to analytic number theory introduction to. From july 11th to july 22nd, 2005, a nato advanced study institute, as part of the series seminaire. Pdf doc del clipboard journal article mr2508649katz, nicholas m. There were about one hundred participants from sixteen countries around the world. Arithmetic randonn ee an introduction to probabilistic. Why the irs cares about the riemann zeta function and. The proof makes use of the following elementary criterium for equidistribution.
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 in number theory, an introduction nato. 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. 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. Introduction theory of benfords law applications conclusions refs. The questions that we discuss are generalizations of the twin prime conjecture. 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. The motivation for this question comes from two different sources. Proceedings of the nato advanced study institute on equidistribution in number theory, montreal, canada, july 1122, 2005.
The point of these notes is not in t he rst place to serve as. We use fourieranalytic methods to give a new proof of bilus theorem on the complex equidistribution of small points on the onedimensional algebraic torus. For an explanation of the background pattern, skip ahead to the end of the page. 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. Pgl 2 acts on the latter by linear change of variables, twisted by inverse determinant. 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. Main equidistribution in number theory, an introduction.
On the expansions of a real number to two distinct bases. With more than 2,200 courses available, ocw is delivering on the promise of open sharing of knowledge. An elementary proof for the equidistribution theorem the mathematical intelligencer september 2015, volume 37, issue 3, pp 12. Equidistribution in number theory, an introduction andrew. 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. There were about one hundred participants from sixteen. 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. Why the irs cares about the riemann zeta function and number theory and why you should too. Equidistribution of primitive rational points on expanding. 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. Equidistribution and benfords law equidistribution yn. 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.
Introduction to analytic number theory introduction to exponential sums. 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. Download it once and read it on your kindle device, pc, phones or tablets. Introduction to analytic number theory fall 2015 if you find a mistake, omission, etc. Equidistribution in number theory, an introduction springerlink. His interest in mathematics is wideranging, and his research focuses on the theory of zeta functions and.
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. We confirm a conjecture of marklof regarding the limiting distribution of certain sparse collections of points on expanding horospheres. Ergodic theoretic proof of equidistribution of hecke. Plots in cof the values of generalized kloosterman sums. Weyl, a doyen of early twentieth century mathematics, presented in. 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. Written for graduate students and researchers alike, this set of lectures provides a structured introduction to the concept of equidistribution in number theory. 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. Equidistribution in number theory, an introduction nato science series ii. First of all, equidistribution is neither a subarea of number theory or analysis nor a technique. Book 237 kindle edition by andrew granville, zeev rudnick. I amaiming tocover basicresultslikelinniksanddukestheorems,aswellascertainaspectsof. An introduction andrew granville, andrew granville, zeav rudnick.
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