On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for \(L\)-functions in the Selberg class (Q2822143)

In this paper, the authors focus on the variances of sums in which the von Mangoldt function is multiplied by arithmetic functions associated with other \(L\)-functions from the Selberg class \({\mathcal S}\), and the sum runs over prime arguments in short intervals. Also, the pair correlation of zeros of \(L\)-functions from the class \({\mathcal S}\) is studied. The authors give a precise computation of ratio and pair correlation of zeros. The obtained results are interesting from the viewpoint that the proofs are based on the pair correlation of the zeros, while an analogue of Hardy-Littlewood conjecture for the autocorrelation of arithmetic functions does not hold in general.NEWLINENEWLINEThis paper is an extension of works by \textit{D. A. Goldston} and \textit{H. L. Montgomery} [Prog. Math. 70, 183--203 (1987; Zbl 0629.10032)], as well as \textit{H. L.~Montgomery} and \textit{K.~Soundararajan} [Commun. Math. Phys. 252, No. 1--3, 589--617 (2004; Zbl 1124.11048)].