On the second moment for primes in an arithmetic progression (Q2759148)

Let NEWLINE\[NEWLINEI(x,h,q,a)= \int_x^{2x} \biggl(\psi (y+h;q,a)- \psi(y;q,a)- \frac{h} {\varphi(q)} \biggr)^2 \,dy.NEWLINE\]NEWLINE The principal result of this paper is the lower bound NEWLINE\[NEWLINEI(x,h,q,a)\geq \biggl( \frac{1}{2}+ o(1)\biggr) \frac{xh}{\varphi(q)} \log \biggl( \frac{xq}{h^3} \biggr),NEWLINE\]NEWLINE valid for \(q\leq h\leq (xq)^{1/3- \varepsilon}\). Unconditionally such a bound is known only for \(q\leq (\log x)^{1-\delta}\), \(h\leq (\log x)^A\) [see \textit{A. E. Özlük}, Bull. Tech. Univ. Istanbul 40, 255--264 (1987; Zbl 0643.10035)]. These results show that the primes cannot be too well distributed in short arithmetic progressions. For comparison, one may conjecture that \(I(x,h,q,a)\sim \frac{xh}{\varphi(q)} \log \frac{xq}{h}\). The argument is based on that used in the first author's work [\textit{D. A. Goldston}, Exp. Math. 13, 366--376 (1995; Zbl 0854.11044)].