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

@inproceedings{Harman1988MetricDA,
  title={Metric diophantine approximation with two restricted variables. III. Two prime numbers},
  author={Glyn Harman},
  year={1988}
}
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.