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Statement

Theorem 1.5 (manuscript p. 2). For every integer $a$,

$$\#\{n\leq x:\sigma(n)\equiv a\pmod n\}=O\!\left(\frac{x}{\log x}\right),$$

and the bound is uniform in $a$: the implied constant does not depend on $a$.

The authors present it as making uniform an upper bound of Pomerance (Acta Arith. 26 (1975), the paper's [15]). They remark (p. 2) that Corollary 3 of [15] appears to give a uniform upper bound, but that its dependence on $a$ is suppressed in the notation.

Source. Paul Pollack, Carl Pomerance, and Lola Thompson, Divisor-Sum Fibers, Mathematika 64(2) (2018), 330--342, DOI 10.1112/S0025579317000535. Theorem 1.5 is on p. 2 of the 11-page author manuscript that the source card identifies.

Read depth. Claims checked: the statement was read clause by clause against the manuscript. The paper gives only a proof sketch (pp. 9--10), which was read for its structure only, not verified.

Proof pointer

"Proof Sketch of Theorem 1.5", end of Section 4, pp. 9--10. Since $P(n)$, the largest prime factor of $n$, divides $n$, it suffices to bound the $n\leq x$ with $P(n)\mid\sigma(n)-a$. Standard estimates discard $O(x/\log x)$ of the $n\leq x$, leaving those with $n>x/\log x$, $P(n)>x^{1/\log\log x}$, and no proper power above $\log^2x$ dividing $n$. Writing $n=pm$ with $p=P(n)$ gives $\sigma(m)\equiv a\pmod p$, display (4.4). When $p>x^{1/2}\log x$, the solutions $m$ for a given $p$ share one value of $\sigma(m)$, and a uniform bound on the number of $m\leq y$ with $\sigma(m)=c$ is summed over the ranges $x/e^{j+1}<p\leq x/e^j$. When $p\leq x^{1/2}\log x$, smooth-number estimates allow $m=uq$ with $q=P(m)>\log x$, and congruence (4.5) determines $q$ from $u$ and $p$. This is a map of the sketch, not a reconstruction of it.

Dependencies

The $\sigma$-analogue of Pomerance's bound on the number of $m\leq y$ with $\varphi(m)=c$, uniform in $c$ (Mathematika 27 (1980) and the 1989 survey Two methods in elementary analytic number theory, the paper's [16] and [17]); standard estimates for smooth numbers.

Bears on

No problem in the corpus. Section 4, where the theorem is proved, concerns the equation $\sigma(n)=kn+a$, and no problem page uses this bound.