We show that for any sequence f:N→{−1,+1} taking values in {−1,+1}, the discrepancy
supn,d∈N∣∣∣∣∑j=1nf(jd)∣∣∣∣
of f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space.
The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

Kaisa Matomäki, Maksym Radziwiłł, 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 p…

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x,x+ψ(x)] with ψ(x)→∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of xϵ-smooth numbers in intervals of the form [x,x+c(ε)x−−√], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n+1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ>0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

Using crowd-sourced and traditional mathematics research, Terence Tao has devised a solution to a long-standing problem posed by the legendary Paul Erdős.

In the middle of the lecture last night, I was thinking to myself that this problem seems like a mixture of combinatorics, integer partitions and coding theory. Something about this article reminds me of that fact again. Most of the references I’m seeing however are directly to number theory and don’t relate to the integer partition piece–perhaps worth delving into to see what shakes out.

The article does a reasonable job of laying out some of the problem and Tao’s solution to it. I was a bit bothered by the idea of “magical” in the title, but it turns out it’s a different reference than the one I was expecting.

Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.

Lecture notes

For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)

We’ve covered the following topics:

Class Introduction

Erdős Discrepancy Problem

n-cubes

Hilbert’s Cube Lemma (1892)

Schur (1916)

Van der Waerden (1927)

Sylvester’s Line Problem (partial coverage to be finished in the next lecture)

Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.

Course Description

Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.

Suggested Prerequisites

Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.

I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.

Category Theory for Winter 2019

As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!