The second moment
is investigated with the novel approximation
and it is shown that when , this leads to more precise conclusions than in the classical Montgomery-Hooley case.
R. C. Vaughan. "Moments for primes in arithmetic progressions, I." Duke Math. J. 120 (2) 371 - 383, 1 November 2003. https://doi.org/10.1215/S0012-7094-03-12026-8