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Statement

Notation. $\phi$ is Euler's totient function and $\sigma$ the sum-of-divisors function. An integer $n$ is a common value of $\phi$ and $\sigma$ when $n=\phi(a)$ and $n=\sigma(b)$ for some positive integers $a,b$.

Theorem 1 (p. 2, quoted). "The equation $\phi(a)=\sigma(b)$ has infinitely many solutions. Moreover, for some positive $\alpha$ and all large $x$, there are at least $\exp\left((\log\log x)^{\alpha}\right)$ integers $n\leqslant x$ which are common values of $\phi$ and $\sigma$."

The proof is unconditional, and the paper says its methods are completely effective (p. 2): the constants are effectively computable (p. 3).

Proof pointer

Section 3, pp. 6--8. The certificate that a $\sigma$-value is a $\phi$-value is the implication (1.1) (p. 2): if $\phi(\operatorname{rad}(m))\mid m$, where $\operatorname{rad}(m)$ is the product of the distinct primes dividing $m$, then $m=\phi\bigl(m\operatorname{rad}(m)/\phi(\operatorname{rad}(m))\bigr)$. The proof splits on whether $x$ is $(\alpha,\varepsilon)$-good, that is, whether the character sums $\Psi(x;m)$ are small for all moduli $3\le m\le x^{\alpha}$ (p. 4); the paper glosses this as, roughly, the absence of the exceptional modulus of Lemma 2.4.

• If $x$ is not good, an exceptional zero exists, and Heath-Brown's theorem (Lemma 2.3, p. 4) supplies many twin primes $p,p+2$ up to $z=m^{500}$. For a set $\mathcal M$ of such primes, each with a distinct large prime factor of $p+1$, the number $\prod_{p\in\mathcal M}(p+1)$ equals both $\sigma\bigl(\prod p\bigr)$ and $\phi\bigl(\prod(p+2)\bigr)$, which gives at least $\exp(z^{\theta/2})$ distinct common values up to $e^{x}$ (pp. 6--7).
• If $x$ is good, the paper takes the primes $p\le x$ with $p+1$ free of prime factors above $x^{1/2-\delta}$ and of every prime in a prime chain (a sequence $q=t_0,t_1,\ldots$ of primes with $t_{j+1}\equiv1\pmod{t_j}$) starting at a prime $q$ from an exceptional set controlled by Lemma 2.6. The Ford--Konyagin--Luca bound on prime chains (Lemma 3.1, p. 6) keeps these removals small. Each $n_j$, the product of $p+1$ over this set with one prime $p_j$ left out, is a $\sigma$-value, and comparing $q$-adic valuations ((3.5) and (3.6), pp. 7--8) shows that (1.1) applies, so $n_j$ is a $\phi$-value. This gives at least $(\gamma/3)x/\log x$ distinct common values below $e^{2x}$ (p. 8).

Either case gives the count in the theorem.

Read depth

Claims checked: Theorem 1, the implication (1.1) and the two cases of the proof were read clause by clause on the page images of the print; the estimates of Section 2 were read for their statements. Lemma 3.1 and Lemma 2.3 are cited from other papers and were not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the Ford--Konyagin--Luca bound on prime chains (its reference [14], Theorem 5), Heath-Brown's theorem that an exceptional zero yields many twin primes (its reference [20], Corollary 2), and classical estimates for primes in progressions (Davenport, Multiplicative number theory).

Source. K. Ford, F. Luca and C. Pomerance, Common values of the arithmetic functions $\phi$ and $\sigma$, Bull. Lond. Math. Soc. 42 (2010), no. 3, 478--488, doi:10.1112/blms/bdq014; pages are those of the edition named on the source card.

Bears on

• Problem 48: the first sentence of Theorem 1 answers the problem's question yes; its solutions $\phi(a)=\sigma(b)$ are pairs of the kind the problem asks for.