Tag - Distribution of primes

Harald Helfgott: Expansion, Divisibility and Parity

5th April, 2024

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for λ the Liouville function (that is, the completely multiplicative function with λ(p) = −1 for every prime),

(1/log x) ∑n ≤ x λ(n) λ(n+1)/n=O(1/√(log log x)),

which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that λ(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Ω(n)=k, for any "popular" value of k (that is, k = log log N+O(√(log log N)) for n ≤ N).

Miguel Walsh: Fourier Uniformity of Multiplicative Functions

25th March, 2024

The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of prime numbers and with other central problems about the behaviour of multiplicative functions, such as the Chowla and Sarnak conjectures.

Jared Duker Lichtman: Twin primes & a modified linear sieve

6th October, 2021

The linear sieve is a powerful tool to tackle problems related to the primes, when combined with equidistribution estimates for the remainder. In 1977 Iwaniec introduced a well-factorable modification of the linear sieve to prove there are infinitely many integers n such that n2+1 has at most two prime factors. Furthermore, the (well-factorable) linear sieve leads to the best known upper bounds for twin primes. These bounds use work of Bombieri, Friedlander, and Iwaniec from 1986, showing these sieve weights equidistribute primes of size x in arithmetic progressions to moduli up to x4/7. This level was recently increased to x7/12 by Maynard.We introduce a new modification of the linear sieve whose weights equidistribute primes of size x to level x10/17. As an application we refine a 2004 upper bound for twin primes of Wu, which gives the largest percent improvement since the work of Bombieri, Friedlander, and Iwaniec.

Allysa Lumley: Primes in short intervals – Heuristics and calculations

17th May, 2021

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length y around x, where y ≪ (log x)2. In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from log x to (log x)2. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.