Skip to main content
edited tags
Link
YCor
  • 63.9k
  • 5
  • 187
  • 286
Source Link
Daniel Loughran
  • 21.3k
  • 3
  • 45
  • 135

Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes.

Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then is the sequence $\{ \alpha p\}$ equidistributed in $[0,1]$, as $p$ runs over primes with $p \equiv a \bmod q$?

Almost certainly this must be known. So I'm looking for a precise reference in the literature as I need it in a paper. Ideally, it would be nice to have an effective version which makes explicit the speed of convergence (via the Erdős-Turán inequality, say).