On the Selberg Integral of the $k$-divisor Function and the $2k$-th Moment of the Riemann Zeta-function

Giovanni Coppola

In the literature one can find links between the $2k$-th moment of the Riemann zeta-function and averages involving $d_k(n)$, the divisor function generated by $\zeta^k(s)$. There are, in fact, two bounds: one for the $2k$-th moment of $\zeta(s)$ coming from a simple average of correlations of the $d_k$; and the other, which is a more recent approach, for the Selberg integral involving $d_k(n)$, 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 $d_k(n)$.