An invitation to additive prime number theory

We survey some of the ideas behind the recent developments in additive number theory, combinatorics and ergodic theory leading to the proof of Hardy- Littlewood type estimates for the number of prime solutions to systems of linear equations of finite complexity.

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest.

In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.

We survey the classical results on the prime number theorem

Ingham studied two types of convolution sums of the divisor function, namely the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their asymptotic formulas as $N \to \infty$. There have been numerous works extending Ingham's work on this convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result Ingham to der...

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.

The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic number theory. The resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: Counting primes in short intervals and in arithmetic progressions; Chowla's conjectu...

The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and examples are considered.

We present a new, elementary, dynamical proof of the prime number theorem.

This paper describes some of the ideas used in the development of our work on small gaps between primes.