On the distribution of primes in intervals of length \(\log^ \Theta N\) (Q1912684)

The authors study relations between the Selberg integral \[ J(N, H)= \int_N^{2N} |\psi(x+ H)- \psi (x)- H|^2 \,dx \] and the square mean \[ I(N, T)= \int_T^{2T} |R(x, T)|^2 \,dx, \] where \(\psi (x)= \sum_{n\leq x} \Lambda (n)\) is the well-known prime counting function, \[ R(x, T)= \psi (x)- x- \sum_{|\gamma |\leq T} w\left({\gamma \over T}\right) {x^\rho \over \rho} \] and \(w(u) =1\) if \(0\leq u\leq 1/2\) and \(w(u)= 2(1- u)\) if \(1/2\leq u\leq 1\). The problem was previously studied by \textit{A. Perelli} and the reviewer [J. Math. Soc. Japan 45, 447--458 (1993; Zbl 0787.11038) and Boll. Unione Mat. Ital., VII. Ser., B 10, 51--66 (1996; Zbl 0856.11040)], who for this purpose introduced \(R(x,T)\) as the remainder term in a modified Riemann-von Mangoldt explicit formula. It was proved that nontrivial estimates of \(J(N, H)\) and \(I(N,T)\) are essentially equivalent when \(H\geq N^\varepsilon\), \(\varepsilon> 0\), and \(T\) is around \(N/H\). The present authors prove that the same is true for \(H=(\log N)^A\) with \(A>1\). (The exact formulation of their results is too involved to be reproduced here).