The binary Goldbach problem with arithmetic weights attached to one of the variables
The Bounded Gap Conjecture and bounds between consecutive Goldbach numbers
The Bouniakowsky conjecture and the density of polynomial roots to prime moduli
The Chen primes contain arbitrarily long arithmetic progressions
The circle and divisor problem
The circle method and pairs of quadratic forms
We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.
The composition of the gcd and certain arithmetic functions.
The Continuity of the Limiting Distribution of a Function of Two Additive Functions.
The cyclic behavior of primes in the arithmetic progressions modulo 11.
The density of representation degrees
For a group and a positive real number , define to be the number of integers less than which are dimensions of irreducible complex representations of . We study the asymptotics of for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups in characteristic zero, showing that either there exists such that for all large , or is virtually abelian (in which case is bounded).
The Dirichlet series of : On an error term associated with its coefficients
The distribution and moments of the error term in the Dirichlet divisor problem
The Distribution of 4-Full Numbers.
The distribution of 4-full numbers
The distribution of additive functions defined on short intervals
The distribution of Farey numbers.
The distribution of f(d) (mod 1)
The distribution of group structures on elliptic curves over finite prime fields.
The Distribution of Integers n Divisible by Lomega(n)
The distribution of inverses modulo a prime in short intervals