Generations of Correlation Averages

We give a general link between weighted Selberg integrals of any arithmetic function and averages of correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions , gauged by the Cesaro weight in the short interval and improved by these some recent results by Ivi？. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2 th moments of Riemann zeta function on the critical line are derived. We also give general requirements on that allow our treatment for weighted Selberg integrals. 1. Introduction and Statement of the Results In the milestone paper [1], Selberg introduced an important tool in the study of the distribution of prime numbers in short intervals with as , that is, the integral where is the von Mangoldt function: if for some prime and ; otherwise, . Thus, is a weighted characteristic function of prime numbers generated by (hereafter, is the Riemann zeta function). The Selberg integral, being a quadratic mean, pertains precisely to the study of the distribution of primes in almost all short intervals , that is, with at most exceptional integers as . Here, we define the Selberg integral of any as where means and is the expected mean value of in short intervals (abbreviated as s.i. mean value). In order to avoid trivialities, we assume that goes to infinity with . In view of nontrivial bounds, the discrete version may be considered close enough to the original continuous integral; so, we are legitimate to use the same symbol for both. Similar considerations hold for the we work with, mostly the -divisor function for , where is the number of ways to write as a product of positive integers (see [2] and compare Section 4). Let us denote the Selberg integral of as with the s.i. mean value of given by where is the residual polynomial of degree such that . The first author [2] has proved the lower bound for , where hereafter. We formulate, for the so-called modified Selberg integral of , where is the same s.i. mean value as in , the following conjecture. In Section 7, Propositions 16 and 18 justify the admissibility of such a choice for the s.i. mean value in modified Selberg integrals, even in the generalization for any divisor function (everything comes from Proposition 14; see Section 7). Hereafter, we abbreviate whenever , for all . Conjecture CL. If , then . A first consequence of our conjecture is the following result. Theorem 1. If