The asymptotic distribution of exponential sums. II.
The average of the least primitive root modulo
The average order of a class of arithmetic functions over arithmetic progressions with applications to quadratic forms.
The average order of a permutation.
The average order of over integers free of large prime factors
The average order of the Dirichlet series of the gcd-sum function.
The average order of two arithmetical functions
The average orders of Hooley’s -functions, II
The Billiard in a Regular Polygon.
The binary additive divisor problem
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).