On the geometric mean of the values of positive multiplicative arithmetical functions (Q6585103)

In the paper under review, the authors investigate the geometric mean \(G_f(n)\) of certain strongly multiplicative positive functions \(f\), given by \N\[\NG_f(n) := \left( \prod_{k=1}^n f(k) \right)^{1/n}. \N\]\NIt is shown that, for strongly multiplicative positive functions \(f\) which are quasi-monomials over primes, i.e., \(f(p) = \alpha(d) p^{d} + O \left(p^{d - \delta} \right) \) for some \(d \in \mathbb{R}\) and \(\alpha(d),\delta >0\), then, for all \(N \in \mathbb{Z}_{\geqslant 1}\), there exist computable constants \(c_1,\dotsc,c_N\) such that \N\[\NG_f(n) = \alpha(d)^{M} e^{d(\gamma + E-1)} \rho_f n^d (\log n)^{\log \alpha(d)} \left( 1 + \sum_{j=1}^N \frac{c_j}{(\log n)^j} + O \left( \frac{1}{(\log n)^{N+1}}\right) \right),\N\]\Nwhere \(\gamma \approx 0.57721 \dotsc\) is the Euler-Mascheroni constant, \(M \approx 0.26149 \dotsc\) is the Meissel-Mertens constant, \(E \approx -1.33258 \dotsc\) is the second Mertens constant, and \N\[\N\rho_f := \prod_p \left( \frac{f(p)}{\alpha(d) p^d}\right)^{1/p}.\N\]\NSome examples are then given. The proof rests on a precise estimate of the sum \N\[\N\sum_{p \leqslant n} \left \lfloor \frac{n}{p} \right \rfloor \log Q(p)\N\]\Nwhere \(Q(x) := \alpha (d) x^d + R(x)\) with \(R(x) \ll x^{d - \delta}\).