An invitation to additive prime number theory

This is a survey of open problems in different parts of combinatorial and additive number theory. The paper is based on lectures at the Centre de Recerca Matematica in Barcelona on January 23 and January 25, 2008.

This is a survey of some of Erd\H os's work on bases in additive number theory.

In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.

For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem...

The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate new arithmetic functions by combining the values of an existing function under an additive operation. The resulting framework not only extends our understanding of classical arithmetic functions but also provides a versatile tool for explo...

Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime number theory, their generalizations, and an extensive list of references on both historical and current progresses.

This is a survey of old and new problems and results in additive number theory.

This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with T.Tao. Contents: 1. An exposition, without much detail, of the work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed discussion of the work of Mauduit and Rivat establishing that 50 percent of the primes have odd digit...

In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.

A survey paper on some recent results on additive problems with prime powers.