The Robin inequality for 7-free integers (Q2882926)

Let \(\Psi_t(n)= n\prod_{p|n} (1+{1\over p}+\cdots+{1\over p^{t-1}})\) for any \(t\geq 2\), so \(\Psi_2(n)\) is the Dedekind function. Let \(R_t(n)= {\Psi_t(n)\over n\log\log n}\) and define the primorial number \(N_n\) of index \(n\) to be the product of the first \(n\) primes. The integer \(n\) is \(t\)-free if, for every prime \(p\), \(p^t\nmid n\).NEWLINENEWLINE The main aim of this paper is to prove that all 7-free integers satisfy Robin's inequality NEWLINE\[NEWLINE\sigma(N)< e^\gamma N\log\log N.NEWLINE\]NEWLINE Since \(\sigma(N)\leq\Psi_7(N)\) for all 7-free integers \(N\), this result is established by obtaining an upper bound for \({\Psi_t(N_n)\over N_n}\) in terms of the \(n\)th prime \(p_n\) in Proposition 4, and then deducing that \(R_t(N)< e^\gamma\) for all \(N\geq N_n\) when \(n\geq n_1(t)\) for a specified \(n_1(t)\). \textit{G. Robin's} inequality for the remaining 7-free \(N< N_{n_1(t)}\) follows from results in [J. Math. Pures Appl. (9) 63, 187--213 (1984; Zbl 0516.10036)] and [Exp. Math. 15, No. 2, 251--256 (2006; Zbl 1149.11041)] by \textit{K. Briggs}.NEWLINENEWLINE The authors also show in Theorem 12 that a sequence \(S_n\) of \(t\)-free integers with \(S_n\geq N_n\) for large enough \(n\) and \(t= o(\log\log n)\) also satisfies Robin's inequality.