Remarks on Fibers of the Sum-of-Divisors Function [chapter]

Paul Pollack
2015 Analytic Number Theory  
Let σ denote the usual sum-of-divisors function. We show that every positive real number can be approximated arbitrarily closely by a fraction m/n with σ (m) = σ (n). This answers in the affirmative a question of Erdős. We also show that for almost all of the elements v of σ (N), the members of the fiber σ −1 (v) all share the same largest prime factor. We describe an application of the second result to the theory of L. E. Dickson's amicable tuples, which are a generalization of the ancient notion of an amicable pair.
doi:10.1007/978-3-319-22240-0_18 fatcat:3dasqsstsneapc6rst6iqe4sjm