Divisor-sum Fibers

نویسنده

  • PAUL POLLACK
چکیده

Let s(·) denote the sum-of-proper-divisors function, that is, s(n) = ∑ d|n, d<n d. Erdős–Granville–Pomerance–Spiro conjectured that for any set A of asymptotic density zero, the preimage set s−1(A ) also has density zero. We prove a weak form of this conjecture: If (x) is any function tending to 0 as x→∞, and A is a set of integers of cardinality at most x 1 2 + , then the number of integers n ≤ x with s(n) ∈ A is o(x), as x → ∞. In particular, the EGPS conjecture holds for infinite sets with counting function O(x 1 2 + ). We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers α and , there are integers n with arbitrarily many s-preimages lying between α(1− )n and α(1 + )n. Finally, we make some remarks on solutions n to congruences of the form σ(n) ≡ a (mod n), proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions n ≤ x, making it uniform in a.

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تاریخ انتشار 2017