The Robin Inequality for 7-Free Integers

Abstract. Recall that an integer is -free if and only if it is not divisible by for some prime . We give a method to check Robin's inequality, , for -free integers and apply it for . We introduce , a generalization of the Dedekind function defined for any integer , by If is -free, then the sum of divisor function is . We characterize the champions for , as primorial numbers. Define the ratio . We prove that, for all , there exists an integer such that we have for , where . Further, by combinatorial arguments, this can be extended to for all such that . This yields Robin's inequality for . For varying slowly with , we also derive .