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DIVISOR-SUM FIBERS

Paul Pollack, Carl Pomerance, Lola Thompson

2018
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Mathematika
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Let $s(\cdot )$ denote the sum-of-proper-divisors function, that is, $s(n)=\sum _{d\mid n,~d<n}d$ . Erdős, Granville, Pomerance, and Spiro conjectured that for any set $\mathscr{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathscr{A})$ also has density zero. We prove a weak form of this conjecture: if $\unicode[STIX]{x1D716}(x)$ is any function tending to $0$ as $x\rightarrow \infty$ , and $\mathscr{A}$ is a set of integers of cardinality at most $x^{1/2+\unicode[STIX]{x1D716}(x)}$
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... , then the number of integers $n\leqslant x$ with $s(n)\in \mathscr{A}$ is $o(x)$ , as $x\rightarrow \infty$ . In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{1/2+\unicode[STIX]{x1D716}(x)})$ . We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D716}$ , there are integers $n$ with arbitrarily many $s$ -preimages lying between $\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D716})n$ and $\unicode[STIX]{x1D6FC}(1+\unicode[STIX]{x1D716})n$ . Finally, we make some remarks on solutions $n$ to congruences of the form $\unicode[STIX]{x1D70E}(n)\equiv a~(\text{mod}~n)$ , 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 $n\leqslant x$ , making it uniform in $a$ .

doi:10.1112/s0025579317000535
fatcat:vcnu5y5rqjbkte2irwp4rvhofe