Robin inequality for $7-$free integers

TL;DR

This paper introduces a new method to verify Robin's inequality for $t$-free integers, focusing on $t=6,7$, by generalizing the Dedekind $ ext{ extPsi}$ function and analyzing its properties for these cases.

Contribution

The authors develop a novel approach using a generalized Dedekind $ ext{ extPsi}_t$ function to check Robin's inequality for specific $t$-free integers, extending the known cases.

Findings

Proved Robin inequality for $t=6,7$.

Characterized champions for the ratio $ ext{ extPsi}_t(n)/n$ as primorial numbers.

Established bounds for the ratio $R_t(n)$ related to Robin's inequality.

Abstract

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $\sigma(n) < e^\gamma n\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $\Psi_t,$ a generalization of Dedekind $\Psi$ function defined for any integer $t\ge 2$ by $$\Psi_t(n):=n\prod_{p | n}(1+1/p+...+1/p^{t-1}).$$ If $n$ is $t-$free then the sum of divisor function $\sigma(n)$ is $ \le \Psi_t(n).$ We characterize the champions for $x \mapsto \Psi_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\frac{\Psi_t(n)}{n\log\log n}.$ We prove that, for all $t$, there exists an integer $n_1(t),$ such that we have $R_t(N_n)< e^\gamma$ for $n\ge n_1,$ where $N_n=\prod_{k=1}^np_k.$ Further, by combinatorial arguments, this can be extended to $R_t(N)\le e^\gamma$ for all $N\ge N_n,$ such that $n\ge n_1(t).$ This yields Robin inequality for $t=6,\,7.$ For $t$ varying slowly with $N$, we also derive $R_t(N)< e^\gamma.$