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arXiv 1706.03120math.NT
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Divisor -总和纤维

Divisor-sum fibers

Paul Pollack, Carl Pomerance, Lola Thompson

发表信息：
Mathematika 64 (2018) 330-342
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摘要：

令$s(\cdot)$表示原因子和函数，即$s(n) = \sum_{d\mid n,~d<n}d$。Erdens-Granville-Pomerance-Spiro证明，对于任何密度为零的集合$\mathcal{A}$，原像集$s^{-1}(\mathcal{A})$也有密度为零。本文证明了这个猜想的一个弱形式：如果$ε(x)$是任何趋于$0$的函数，且$\mathcal{A}$是基数至多为$x^{\frac12+ε(x)}$的整数集合，则具有$s(n) \in \mathcal{A}$的整数$n\le x$的个数是$o(x)$，as $x\to\infty$ .特别地，EGPS猜想对于具有计数函数$O(x^{\frac12 + ε(x)})$的无限集合成立。我们还反驳了一个假设从同一文件的EGPS表明，任何正数$α$和$ε$，有整数$n$与任意多个$s$ -原像之间的$α(1-ε)n$和$α(1+ε)n$。最后，我们对解$n$到$σ(n) \equiv a\pmod{n}$形式的同余作了一些评论，提出了对前两位作者最近工作中出现的一个猜想的修正。我们还改进了以前的上限的解决方案$n \leq x$的数量，使其均匀的$a$。

英文摘要：

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erdős-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 $ε(x)$ is any function tending to $0$ as $x\to\infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac12+ε(x)}$, then the number of integers $n\le x$ with $s(n) \in \mathcal{A}$ is $o(x)$, as $x\to\infty$. In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{\frac12 + ε(x)})$. 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 $σ(n) \equiv a\pmod{n}$, 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 $a$.

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