Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals (Q2755158)

The main aim of this paper is the evaluation of NEWLINE\[NEWLINEI(\sigma, T) := \int_1^T\left|{\zeta'\over\zeta}(\sigma + it)\right|^2 dt, NEWLINE\]NEWLINE where throughout the paper the authors assume the truth of the Riemann Hypothesis (RH) that all complex zeros of the Riemann zeta-function \(\zeta(s)\) have real parts equal to 1/2, so that all the results of the paper are conditional. NEWLINENEWLINENEWLINEIn Theorem 1, the authors prove an asymptotic formula for \(I(\sigma,T)\), assuming that \(\sigma = {1\over 2} + {a\over\log T}\) and \(0 < a \ll 1\). Their expression contains a double sum over ordinates \(\gamma, \gamma'\) of zeros of \(\zeta(s)\). Such sums over the zeros are related to the function \(F(\alpha, T)\) introduced by \textit{H. L. Montgomery} [Proc. Symp. Pure Math. 24, 181-193 (1973; Zbl 0268.10023)], where he made the conjecture (MH) that \(F(\alpha, T) \sim 1\) as \(T\to\infty\) uniformly for bounded \(\alpha\). One of the implications of the MH is Montgomery's so-called pair correlation conjecture about the spacing of the \(\gamma\)'s. In Theorem 3, the authors show the equivalence of three statements (all assuming the RH). As they comment afterwards, statement (B) is an equivalent of the pair correlation conjecture, so that their result in fact gives three equivalents of the pair correlation conjecture. Similarly, their Theorem 4 gives three equivalents of the so-called essential simplicity hypothesis of the zeros of the zeta-function. The subject of the vertical distribution of zeros is naturally connected to problems involving primes in short intervals. Using a corollary of Theorem 1, the authors prove in Theorem 2 the bound NEWLINE\[NEWLINE J(\beta, T) \ll {\beta\log^2T\over T} \qquad(1 \leq \beta\leq T/\log^3T), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE J(\beta, T) = \int_1^{T^\beta}\left(\psi(x+{x\over T}) - \psi(x) - {x\over T}\right)^2{dx\over x^2} NEWLINE\]NEWLINE and \(\psi(x) = \sum_{p^m\leq x}\log p\) is the summatory function of the von Mangoldt function. The proof of the main result, namely Theorem 1, is classical. The partial fraction formula for \(\zeta'(w)/\zeta(w)\) is used, together with the techniques of Montgomery's paper (op. cit.). NEWLINENEWLINENEWLINEThe proof of Theorem 2 uses a lemma of Selberg for a weighted integral of \((\psi(x+{x\over T}) - \psi(x) - {x\over T})^2\), while the proof of Theorem 3 rests on Tauberian arguments contained in Lemmas 2-4. Theorem 4 follows by the same methods used in the proof of Theorem 3.