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- W3100632772 abstract "Let $s(cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = sum_{dmid n,~d<n}d$. ErdH{o}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: If $epsilon(x)$ is any function tending to $0$ as $xtoinfty$, and $mathcal{A}$ is a set of integers of cardinality at most $x^{frac12+epsilon(x)}$, then the number of integers $nle x$ with $s(n) in mathcal{A}$ is $o(x)$, as $xtoinfty$. In particular, 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 $alpha(1+epsilon)n$. Finally, we make some remarks on solutions $n$ to congruences of the form $sigma(n) equiv apmod{n}$, 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 $n leq x$, making it uniform in $a$." @default.
- W3100632772 created "2020-11-23" @default.
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- W3100632772 date "2018-01-01" @default.
- W3100632772 modified "2023-10-17" @default.
- W3100632772 title "DIVISOR‐SUM FIBERS" @default.
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- W3100632772 doi "https://doi.org/10.1112/s0025579317000535" @default.
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