An invitation to additive prime number theory

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs...

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both $\omega$, the number of distinct primes of $n$, and $\Omega$, the total number primes of $n$, are members of $F_0$. As an application of the exponential sum result, we use the Hardy-Littlewood circle method to find the asymptotics of the Goldbac...

A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations of the mentioned problems.

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for it, where $a p$ and $b P_{2}$ are both square-free, $p$ is a prime, and $P_{2}$ has at most two prime factors. We also consider some special cases where $p$ is small, $p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical progressions and a Goldbach-type upper bound result. Our new results generalize and improv...

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more simplified manner. Basic properties of some key concepts such as infinitesimals, the associated nonarchimedean absolute values, invariance of measure and cardinality of a compact subset of the real line under an IFS are discussed more thorou...

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. W...

This is an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers.

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to pro...

The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy-Littlewood method has been generalised to obt...