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Theorem 1 of Garaev states that for every $A>0$ and every $x>x_0(A)$ at least $\exp((\log\log x)^{A})$ integers $n\le x$ are values of both $\phi$ and $\sigma$. Since the count is unbounded in $x$, the equation $\phi(n)=\sigma(m)$ has infinitely many solutions, which answers Problem 48 in the affirmative. The result sharpens Theorem 1 of Ford, Luca and Pomerance (Ford–Luca–Pomerance 2010), which gives the same count for one unspecified exponent $\alpha>0$. Garaev's introduction says that the proof is based on a refined version of Konyagin's approach, the alternative route described in a remark of the earlier paper that avoids Heath-Brown's theorem linking Siegel zeros to twin primes, and on the argument of that paper; Garaev's Lemma 4 follows from the proof of its Lemma 2.6. The common values are again built as $\sigma$ of a product of primes $p\le x$ with $p+1$ smooth and certified as totient values through the implication that $\phi(\operatorname{rad}(m))\mid m$ makes $m$ a value of $\phi$; Lemma 1 uses Landau's theorem on the real zeros of two real primitive $L$-functions, and Lemma 2 produces a scale at which $L(s,\chi)$ has no zeros in a suitable region for all moduli up to a small power of $x$. The digest is on the source card garaev_2011_number_common_values_arithmetic_functions_below.

Depends on. Ford–Luca–Pomerance 2010, whose argument the proof refines and whose Lemma 2.6 gives Garaev's Lemma 4.

Source. M. Z. Garaev, On the number of common values of arithmetic functions $\phi$ and $\sigma$ below $x$, Mosc. J. Comb. Number Theory 1 (2011), no. 3, 42–49, received 17 May 2011. The journal's record gives the volume, the issue and the year and no month or day, so this page is dated by the first day of 2011.

Acceptance. Refereed: the paper is a journal article in the Moscow Journal of Combinatorics and Number Theory, volume 1, issue 3 (2011). Reviewed: the site's curator, Thomas F. Bloom, labels the problem PROVED (LEAN) and credits the improved lower bound on the number of common values to this paper in the problem's commentary (page last edited 2025-10-17, read 2026-10-07); the curator had no part in the result. The site's Lean marker refers to a formalization of the Ford–Luca–Pomerance argument, recorded on that claim page, and not to this paper, so no formalized evidence is listed. Nothing here is this project's own review.