We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.
In this paper we establish the distribution of prime numbers in a given arithmetic progression for which is squarefree.
The dynamical Mertens' theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynamical systems modulo the equivalence of shared orbit-growth asymptotics is not countable. In contrast, for the equivalence relation of measurable isomorphism...