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Statement

For integers $n_1,n_2$ with $0<|n_1-n_2|\le N$, and sufficiently large $N$,

$$\sum_{q\mid\gcd(n_1,n_2)}\frac1q\ll\log\log\log N,$$

where $q$ ranges over prime powers $p^a$ with $a\ge1$.

Source. Bloom, arXiv:2112.03726v2, Lemma 3, pp. 12–13; the paper identifies this as Croot's Lemma 2.

Rewritten proof

Every common divisor divides $m=|n_1-n_2|\in[1,N]$. The contribution of powers of exponent at least two is bounded independently of $N$, since

$$\sum_p\sum_{a\ge2}p^{-a} =\sum_p\frac1{p(p-1)} \le\sum_{j\ge2}\frac1{j(j-1)}=1.$$

The integer $m$ has at most $\log N/\log2$ distinct prime divisors, because the product of $r$ distinct primes is at least $2^r$. A sum of reciprocals of at most this many primes is maximized by the smallest primes. The Chebyshev lower bound $\pi((\log N)^2)\gg(\log N)^2/\log\log N$ exceeds this number for large $N$. Thus

$$\sum_{p\mid m}\frac1p \le\sum_{p\le(\log N)^2}\frac1p \ll\log\log\log N$$

by Mertens' reciprocal-prime estimate. Add the bounded higher-power contribution.

Dependencies

External Mertens and Chebyshev estimates, recorded on pp. 3 and 12–13. This overlap estimate is used by Proposition 3.

Bears on

• Problem 298
• Problem 299