On the Selberg integral of the k-divisor function and the 2k-th moment of the Riemann zeta-function

Abstract

In the literature one can find links between the 2k-th moment of the Riemann zeta-function and averages involving dk(n), the divisor function generated by ?k(s). There are, in fact, two bounds: one for the 2k-th moment of ?(s) coming from a simple average of correlations of the dk; and the other, which is a more recent approach, for the Selberg integral involving dk(n), applying known bounds for the 2k-th moment of the zeta-function. Building on the former work, we apply an elementary approach (based on arithmetic averages) in order to get the reverse link to the second work; i.e., we obtain (conditional) bounds for the 2k-th moment of the zeta-function from the Selberg integral bounds involving dk(n).