DIVISOR-SUM FIBERS. Issue 2 (3rd April 2018) - Pollack, Paul; Pomerance, Carl; Thompson, Lola
Abstract : Let s(· ) denote the sum-of-proper-divisors function, that is, s(n)=_dn, d<nd . Erdős, Granville, Pomerance, and 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 [STIX]x1D716(x) is any function tending to 0 as x→ ∞, and A is a set of integers of cardinality at most x^1/2+[STIX]x1D716(x), then the number of integers nx 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+[STIX]x1D716(x)) . We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers [STIX]x1D6FC and [STIX]x1D716, there are integers n with arbitrarily many s -preimages lying between [STIX]x1D6FC(1-[STIX]x1D716)n and [STIX]x1D6FC(1+[STIX]x1D716)n . Finally, we make some remarks on solutions n to congruences of the form [STIX]x1D70E(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 nx, making it uniform in a .