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 ) = ∑ d ∣ n, d < n d . 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 ε ( x ) is any function tending to 0 as x → ∞, and A is a set of integers of cardinality at most x 1 / 2 + ε ( x ), 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 + ε ( x ) ) . We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers α and ε, there are integers n with arbitrarily many s ‐preimages lying between α ( 1 − ε ) n and α ( 1 + ε ) 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 .