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URL : | http://terrytao.wordpress.com/ | |
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Filed Under: | Academics / Mathematics | |

Posts on Regator: | 488 | |

Posts / Week: | 1.5 | |

Archived Since: | December 12, 2008 |

In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem: Problem 1 (Erdös-Ulam problem) Let be a set such that the distance between any two points in is rational. Is it true that cannot be (topologically) dense in ? The paper of Anning and Erdös addressed the case that […]

Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes“. This is a followup work to our two previous papers (discussed in this previous post), in which we had simultaneously shown that the maximal gap between primes up to exhibited a lower bound […]

In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The […]

In Notes 1, we approached multiplicative number theory (the study of multiplicative functions and their relatives) via elementary methods, in which attention was primarily focused on obtaining asymptotic control on summatory functions and logarithmic sums. Now we turn to the complex approach to multiplicative number theory, in which the focus is instead on obtaining […]

We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. We will not attempt a comprehensive review of this subject; […]

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple eigenvalues“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the […]

Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals, and the classical ring of integers, are placed inside the […]

Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals, and the classical ring of integers, are placed inside the […]

In analytic number theory, an arithmetic function is simply a function from the natural numbers to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than or, as in this previous blog post, but we will restrict attention here to the classical situation of real or […]

Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of (affine-)linear forms, none of which is a multiple of any other, find a number such that a certain property of the linear forms are true. For instance: For the twin prime conjecture, one […]

In the winter quarter (starting January 5) I will be teaching a graduate topics course entitled “An introduction to analytic prime number theory“. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers. I will list the topics I […]

The wave equation is usually expressed in the form where is a function of both time and space, with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding […]

I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue of a prime. Show More Summary

The prime number theorem can be expressed as the assertion as, where is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula where the second von Mangoldt function is defined […]

In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence of finite graphs, one can extract a subsequence which converges (in a specific sense) to […]

One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups). Theorem 1 (Cayley’s theorem) Let be a group of some finite order. Then is isomorphic to a subgroup of the symmetric group on elements. Furthermore, this subgroup […]

The (presumably) final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“. This article, submitted to the Newsletter of the European...Show More Summary

Analytic number theory is often concerned with the asymptotic behaviour of various arithmetic functions: functions or from the natural numbers to the real numbers or complex numbers. In this post, we will focus on the purely algebraic properties of these functions, and for reasons that will become clear later, it will be convenient to […]

Tamar Ziegler and I have just uploaded to the arXiv our paper “Narrow progressions in the primes“, submitted to the special issue “Analytic Number Theory” in honor of the 60th birthday of Helmut Maier. The results here are vaguely reminiscent of the recent progress on bounded gaps in the primes, but uses different methods. About […]

Kevin Ford, Ben Green, Sergei Konyagin, and myself have just posted to the arXiv our preprint “Large gaps between consecutive prime numbers“. This paper concerns the “opposite” problem to that considered by the recently concluded Polymath8 project, which was concerned with very small values of the prime gap. Here, we wish to consider the […]

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