ON THE SELBERG INTEGRAL OF THE $k$-DIVISOR FUNCTION AND THE $2k$-TH MOMENT OF THE RIEMANN ZETA-FUNCTION

Giovanni Coppola

Abstract: 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 $d_k(n)$.

Classification (MSC2000): 11M06; 11N37

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