Glyn Harman

Trigonometric sums over primes I

We write e ( x ) for e 2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean | A | ≤ C | B | where C is a positive constant depending at most on an arbitrary positive number e, and on an integer k . The letter p always denotes a prime number. The main results of the present paper are as follows.

The Brun-Titchmarsh Theorem on average

Throughout this paper a denotes a fixed non-zero integer and the letter p with or without subscript denotes a prime variable. As usual, for (q, a) = 1 we write

Metric diophantine approximation with two restricted variables. III. Two prime numbers

\pi \left( {x;\,q,\,a} \right)\, = \,\sum\limits_{\mathop {p \leqslant x}\limits_{p \equiv a(\bmod \,q)} } {1.}

On the distribution of √ p modulo one

We show how the methods of Vaughan & Wooley, which have proved fruitful in dealing with Waring’s problem, may also be used to investigate the fractional parts of an additive form. Results are obtained which are near to best possible for forms with algebraic coefficients. New results are also obtained in the general case. Extensions are given to make several additive forms simultaneously small. The key ingredients in this work are: mean value theorems for exponential sums, the use of a small common factor for the integer variables, and the large sieve inequality.

The three primes theorem with almost equal summands

Abstract It is shown that, if ψ ( n ) is a real function with 0 1 2 , and satisfies a simple regularity condition, then the inequality | αp − q | ψ ( p ) has infinitely many solutions in primes p and q for almost all α if and only if ∑ n=2 ∞ ψ(n)( log n) −2 = ∞ For example, there are infinitely many solutions in primes when ψ ( n ) = n −1 (log n ) β if and only if β ≥ 1.

On the distribution of ap k modulo one

For the purpose of this paper, we call a set of positive reals ν a well-spaced set if there is a c > 0 such that

Approximation of real matrices by integral matrices

The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more generally, prime ideals in algebraic number fields). One consequence of our main theorem (Theorem 2 below) is the following result on rational primes.