Moments for primes in arithmetic progressions, I

Abstract

The second moment

$${\displaystyle \sum _{q\le Q}{\displaystyle \sum _{a=1}^q{(\psi (x;q,a)-\rho (x;q,a))}^2}}$$

is investigated with the novel approximation

$$\rho (x;q,a)={\displaystyle \sum _{\begin{array}{c}n\le x\\ n\equiv a(\text{mod\hspace{0.17em}}q)\end{array}}{F}_R(n),}$$

where

$$F_R(n)={\displaystyle \sum _{r\le R}\frac{\mu (r)}{\varphi (r)}{\displaystyle \sum _{\begin{array}{c}b=1\\ (b,r)=1\end{array}}^{r}e(bn/r),}}$$

and it is shown that when $R\le {\text{log}}^{A}x$, this leads to more precise conclusions than in the classical Montgomery-Hooley case.