Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:377
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Archived Since:December 12, 2008

Blog Post Archive

Newton iteration and the Siegel linearisation theorem

An extremely large portion of mathematics is concerned with locating solutions to equations such as or for in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps or. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial […]

The ergodic theorem and Gowers-Host-Kra seminorms without separability or amenability

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space, and is a vector in that Hilbert space, then one has in the strong topology, where is the -invariant subspace of, and is the orthogonal projection to. […]

Random matrices: tail bounds for gaps between eigenvalues

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency graph of an Erd\H{o}s-Renyi graph) almost surely had […]

254A, Notes 8: The Hardy-Littlewood circle method and Vinogradov’s theorem

We have seen in previous notes that the operation of forming a Dirichlet series or twisted Dirichlet series is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions and are multiplicative characters. Similarly, it turns out that the operation of forming […]

An averaged form of Chowla’s conjecture

Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that as for any distinct natural numbers, where denotes the Liouville function. (One could also […]

254A, Supplement 7: Normalised limit profiles of the log-magnitude of the Riemann zeta function (optional)

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function in the critical strip in the limit. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude of the zeta function. (In principle, one can reconstruct a […]

254A, Supplement 6: A cheap version of the theorems of Halasz and Matomaki-Radziwill (optional)

In analytic number theory, it is a well-known phenomenon that for many arithmetic functions of interest in number theory, it is significantly easier to estimate logarithmic sums such as than it is to estimate summatory functions such as (Here we are normalising to be roughly constant in size, e.g. as.) For instance, when is […]

254A, Notes 7: Linnik’s theorem on primes in arithmetic progressions

In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime […]

254A, Notes 6: Large values of Dirichlet polynomials, zero density estimates, and primes in short intervals

In the previous set of notes, we studied upper bounds on sums such as for that were valid for all in a given range, such as ; this led in turn to upper bounds on the Riemann zeta for in the same range, and for various choices of. While some improvement over the trivial […]

254A, Notes 5: Bounding exponential sums and the zeta function

We return to the study of the Riemann zeta function, focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of Notes 2, such upper bounds can lead to zero-free regions for, which in turn lead to improved estimates for the […]

254A, Supplement 5: The linear sieve and Chen’s theorem (optional)

We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the linear sieve in which the sieve dimension is equal to, which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. […]

254A, Notes 4: Some sieve theory

Many problems in non-multiplicative prime number theory can be recast as sieving problems. Consider for instance the problem of counting the number of pairs of twin primes contained in for some large ; note that the claim that for arbitrarily large is equivalent to the twin prime conjecture. One can obtain this count by any […]

254A, Notes 3: The large sieve and the Bombieri-Vinogradov theorem

A fundamental and recurring problem in analytic number theory is to demonstrate the presence of cancellation in an oscillating sum, a typical example of which might be a correlation between two arithmetic functions and, which to avoid technicalities we will assume to be finitely supported (or that the variable is localised to a finite […]

254A, Supplement 4: Probabilistic models and heuristics for the primes (optional)

We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins. This creates a major jump in difficulty; for instance, even the […]

The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture

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 […]

Long gaps between primes

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 […]

254A, Supplement 3: The Gamma function and the functional equation (optional)

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 […]

254A, Notes 2: Complex-analytic multiplicative number theory

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 […]

254A, Supplement 2: A little bit of complex and Fourier analysis

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; […]

Random matrices have simple spectrum

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 […]

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