Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:411
Posts / Week:1.1
Archived Since:December 12, 2008

Blog Post Archive

A cheap version of Halasz’s inequality

A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum. Theorem 1 (Halasz inequality) Let be a multiplicative function bounded in magnitude by, and suppose that,, […]

275A, Notes 5: Variants of the central limit theorem

In the previous set of notes we established the central limit theorem, which we formulate here as follows: Theorem 1 (Central limit theorem) Let be iid copies of a real random variable of mean and variance, and write. Then, for any fixed, we have as. This is however not the end […]

Chains of large gaps between primes

Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “Chains of large gaps between primes“. This paper was announced in our previous paper with Konyagin and Green, which was concerned with the largest gap between consecutive primes up to, in which we improved the Rankin bound of to for large […]

Klaus Roth

Klaus Roth, who made fundamental contributions to analytic number theory, died this Tuesday, aged 90. I never met or communicated with Roth personally, but was certainly influenced by his work; he wrote relatively few papers, but they tended to have outsized impact. For instance, he was one of the key people (together with Bombieri) to […]

Analytic number theory program at MSRI: Jan-May 2017 (second announcement)

Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the […]

275A, Notes 4: The central limit theorem

Let be iid copies of an absolutely integrable real scalar random variable, and form the partial sums. As we saw in the last set of notes, the law of large numbers ensures that the empirical averages converge (both in probability and almost surely) to a deterministic limit, namely the mean of the reference […]

A small remark on the Elliott conjecture

The Chowla conjecture asserts, among other things, that one has the asymptotic as for any distinct integers, where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts, but for sake of discussion let us focus on the shift case.) This […]

275A, Notes 3: The weak and strong law of large numbers

One of the major activities in probability theory is studying the various statistics that can be produced from a complex system with many components. One of the simplest possible systems one can consider is a finite sequence or an infinite sequence of jointly independent scalar random variables, with the case when the are also identically […]

275A, Notes 2: Product measures and independence

In the previous set of notes, we constructed the measure-theoretic notion of the Lebesgue integral, and used this to set up the probabilistic notion of expectation on a rigorous footing. In this set of notes, we will similarly construct the measure-theoretic concept of a product measure (restricting to the case of probability measures to avoid […]

Sweeping a matrix rotates its graph

I recently learned about a curious operation on square matrices known as sweeping, which is used in numerical linear algebra (particularly in applications to statistics), as a useful and more robust variant of the usual Gaussian elimination operations seen in undergraduate linear algebra courses. Given an matrix (with, say, complex entries) and an index, […]

275A, Notes 1: Integration and expectation

In Notes 0, we introduced the notion of a measure space, which includes as a special case the notion of a probability space. By selecting one such probability space as a sample space, one obtains a model for random events and random variables, with random events being modeled by measurable sets in, and […]

275A, Notes 0: Foundations of probability theory

Starting next week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of Shannon entropy played a crucial role in my recent paper on the Chowla and Elliott conjectures, and random […]

Entropy and rare events

Let and be two random variables taking values in the same (discrete) range, and let be some subset of, which we think of as the set of “bad” outcomes for either or. If and have the same probability distribution, then clearly In particular, if it is rare for to lie in, […]

The logarithmically averaged Chowla and Elliott conjectures for two-point correlations; the Erdos discrepancy problem

I’ve just uploaded two related papers to the arXiv: The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, submitted to Forum of Mathematics, Pi; and The Erdos discrepancy problem, submitted to the new arXiv overlay journal, Discrete Analysis (see this recent announcement on Tim Gowers’ blog). Show More Summary

The Erdos discrepancy problem via the Elliott conjecture

The Chowla conjecture asserts that all non-trivial correlations of the Liouville function are asymptotically negligible; for instance, it asserts that as for any fixed natural number. This conjecture remains open, though there are a number of partial results (e.g. Show More Summary

Sign patterns of the Mobius and Liouville functions

Kaisa Matomaki, Maksym Radziwi??, and I have just uploaded to the arXiv our paper “Sign patterns of the Liouville and Möbius functions“. This paper is somewhat similar to our previous paper in that it is using the recent breakthrough of Matomaki and Radziwi?? on mean values of multiplicative functions to obtain partial results towards the […]

Heath-Brown’s theorem on prime twins and Siegel zeroes

The twin prime conjecture is one of the oldest unsolved problems in analytic number theory. There are several reasons why this conjecture remains out of reach of current techniques, but the most important obstacle is the parity problem which prevents purely sieve-theoretic methods (or many other popular methods in analytic number theory, such as the […]

A wave equation approach to automorphic forms in analytic number theory

The Poincaré upper half-plane (with a boundary consisting of the real line together with the point at infinity ) carries an action of the projective special linear group via fractional linear transformations: Here and in the rest of the post we will abuse notation by identifying elements of the special linear group with their equivalence […]

Equidistribution for multidimensional polynomial phases

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that for any continuous (or equivalently, for any smooth) function. By approximating uniformly by a Fourier series, this claim is equivalent to that of showing that for any non-zero integer (where ), which […]

Analytic Number Theory program at MSRI: Jan-May 2017

Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the […]

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