Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Lola Thompson
Affiliation:

Oberlin College
Date:

Tue, 2018-09-04 09:50 - 10:10 Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) =\sum_{d\mid n,~d<n}d$.

ErdÃ¶s--Granville--Pomerance--Spiro conjectured that, for any set $\mathcal{A}$ of asymptotic

density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak

form of this conjecture. In particular, we show that the EGPS conjecture holds for

infinite sets with counting function $O(x^{\frac12 + \epsilon(x)})$. We also disprove a hypothesis

from the same paper of EGPS by showing that for any positive numbers $\alpha$ and $\epsilon$,

there are integers $n$ with arbitrarily many $s$-preimages lying between $\alpha(1-\epsilon)n$

and $\alpha(1+\epsilon)n$. This talk is based on joint work with Paul Pollack and Carl Pomerance.

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/7866