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On the Selberg integral of the \(k\)-divisor function and the \(2k\)-th moment of the Riemann zeta-function. (English) Zbl 1337.11056

Summary: 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 \(d_k(n)\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N37 Asymptotic results on arithmetic functions
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