Remarks on Fibers of the Sum-of-Divisors Function

Paul Pollack ()
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Paul Pollack: University of Georgia

A chapter in Analytic Number Theory, 2015, pp 305-320 from Springer

Abstract: Abstract Let σ $$\mbox{$\sigma$}$$ 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 ) $$\sigma (m) =\sigma (n)$$ . This answers in the affirmative a question of Erdős. We also show that for almost all of the elements v of σ ( N ) $$\sigma (\mathbf{N})$$ , the members of the fiber σ − 1 ( v ) $$\sigma ^{-1}(v)$$ 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.

Keywords: Amicable Pairs; Largest Prime Factor; Ancient Notion; Recent Spectacular Progress; Prime Divisor (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-22240-0_18

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DOI: 10.1007/978-3-319-22240-0_18

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