Arithmetic progressions and the primes - El Escorial lectures

We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of arithmetic progressions that satisfy additional congruence conditions and provide asymptotic approximations.

In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of th...

An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length.

We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs of Roth's theorem), we do not pass to a progression related to the large Fourier coefficients of our set (as most other proofs of Roth do). Furthermore, in our proof, the density increment is achieved through an application of a quantitativ...

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset of) the primes up to $N$ with common difference $O((\log N)^{L_k})$, for an unspecified constant $L_k$. In this work we obtain this statement with the precise value $L_k = (k-1) 2^{k-2}$. This is achieved by proving a relative version of S...

We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{C}$, we will establish that for any small constant $\epsilon>0$, the set $\left\{q_n| q_{n+1}-q_n \leq \epsilon \log n \right\}$ constitutes a positive proportion of all prime numbers. U...

We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic progressions. More precisely, the set of natural numbers n for which all components n+h_i (i=1,2,...m) are consecutive primes contains arbitrarily long (finite) arithmetic progressions. Moreover, the set of m-tuples that satisfy this property represen...