A lower bound for the second moment of primes in short intervals (Q1906663)

The author and \textit{H. L. Montgomery} [in Analytic number theory and diophantine problems, Birkhäuser, Prog. Math. 70, 183-203 (1987; Zbl 0629.10032)] proved that, under the Riemann hypothesis, the asymptotic equality \[ I(x, h):= \int^x_0 (\psi (t+h)- \psi(t)- h)^2 dt\sim hx\log {\textstyle {x\over h}} \] is equivalent to a suitable form of Montgomery's pair correlation conjecture. Improving on his own work, the author proves the lower bound \(I(x,h)\geq ({1\over 2}- 2\alpha- \varepsilon)hx \log x\) where \(h= x^\alpha\) and \(0\leq \alpha\leq {1\over 4}\), assuming the generalized Riemann hypothesis. His previous result [Q. J. Math., Oxf. II. Ser. 35, 153-163 (1984; Zbl 0532.10024)] was a similar lower bound for the shorter range \(0\leq \alpha\leq {1\over 6}\), under GRH. The method of the proof is similar to the one exploited by the author in several papers and consists of the use of the function \(\lambda_Q (n)= \sum_{q\leq Q} \mu(q) \varphi (q)^{-1} c_q (n)\), where \(c_q (n)\) is the Ramanujan sum. The function \(\lambda_Q\) mimics, in a sense, the behaviour of the \(\Lambda\) functions over arithmetic progressions. The use of that function goes back to \textit{A. Selberg} [Skr. Norske Vid.-Akad., Oslo No. 10 (1943; Zbl 0028.11101)] and it has also been exploited by \textit{D. R. Heath-Brown} [Rev. Mat. Iberoam. 1, 45-59 (1985; Zbl 0599.10041)] in his work on the ternary Goldbach problem.