Arithmetic progressions and the primes - El Escorial lectures

We show estimates for the distribution of $k$-free numbers in short intervals and arithmetic progressions. We argue that, at least in certain ranges, these estimates agree with a conjecture by H. L. Montgomery.

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having density at least s, and having the least number of 3-term arithemtic progressions among all sets of density at least s, then S is nearly translation invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that |S intersect (...

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$ in $A$ with $0 < |r| \leq N^{o(1)}$, if $N$ is sufficiently large depending on $k,P_1,\dots,P_k$ and $\delta$. In this paper we shorten the size of this progression to $0 < |r| \leq ...

We give some corrections of our paper "Primes in arithmetic progressions to large moduli"[BFI]. The corrections do not affect the statements of any of the theorems in the paper. The contents of our two sequel papers [BFI2, BFI3] also remain unchanged.

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of Polymath, who had previously obtained the exponent $\tfrac{1}{2} + \tfrac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that $\liminf_{n \rightarrow \infty} (p_{n+m}-p_n) = O(\ex...

In this work and its sister paper [5] we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dipense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and reelied on in earlier proofs.

This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang \cite{Z} had revived in us an intention to produce a second edition of our book "Opera de Cribro", one which would include an account of Zhang's result, stressing the sieve aspects of the method. A complete and connected version of the proof, in our style but not intended for journal publication, seemed a natural first step in this project. ...

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.

This is the text accompanying my Bourbaki seminar on the work of Bloom and Sisask, Croot, Lev, and Pach, and Ellenberg and Gijswijt.