We give a general link between weighted Selberg integrals of any arithmetic function f and averages of f 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 dk(n), gauged by the Cesaro weight in the short interval n∈x-H,x+H and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2kth moments of Riemann zeta function on the critical line are derived. We also give general requirements on f that allow our treatment for f weighted Selberg integrals.