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Pollack's Theorem 1 states that for every real $\beta>0$ and every $\varepsilon>0$ there are positive integers $m,n$ with $\sigma(m)=\sigma(n)$ and $|m/n-\beta|<\varepsilon$. Taking $\beta=\alpha\ge1$ and $\varepsilon=1/k$ for $k=1,2,\dots$ gives integers $n_k,m_k$ with $\sigma(n_k)=\sigma(m_k)$ and $n_k/m_k\to\alpha$, which is the sequence the problem asks for. The proof shows that the closure of $\{\log(m/n):\sigma(m)=\sigma(n)\}$ is the whole real line; it adapts the Schinzel--Sierpiński construction of equal-$\sigma$ pairs with large ratio, replacing the prime $k$-tuples conjecture with Zhang's bounded-gaps theorem in the form, stated for general admissible linear forms, that Maynard proved. The statement and the paper's other results are on the source card Pollack 2015, which follows the author's manuscript at the second link, from the author's research page; the proof is not reconstructed in this repository.

Acceptance. Reviewed: the site's curator, Thomas F. Bloom, records the answer as yes, proved by Pollack, on the problem page (last edited 2025-09-28). Published: P. Pollack, Remarks on fibers of the sum-of-divisors function, in Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday (Springer, 2015), 305--320; a chapter of an edited volume, so no journal refereeing is listed as evidence. The volume gives the year 2015 and no day; the page's date is the day the chapter's DOI record was created, 2015-11-18. The site's remarks and Erdős's 1974 paper [Er74b], on the card Erdős 1974, call the analogous statement for Euler's totient easy, and Pollack's Remark 3 says that the argument, with obvious modifications, proves Theorem 1 with $\varphi$ in place of $\sigma$.