Divisors in windows and the Euler--Mascheroni constant

Abstract

For $0<c<1$ the number of divisors of $q$ below $q^{c}$ averages, over $q\le B$, to $c\log B+(γ-c)+O(B^{-\min(c,1-c)})$, where $γ$ denotes the Euler--Mascheroni constant. Consequently the number of divisors in a window $(q^{a},q^{b})$ averages to $(b-a)(\log B-1)+o(1)$: the constant $γ$ cancels, and the average counts for two windows stand asymptotically in the ratio of the windows' exponent lengths. In particular a number has, on average, twice as many divisors in $(n^{1/4},n^{1/2})$ as in $(n^{1/8},n^{1/4})$, by an argument nowhere mentioning $γ$. Anchoring a window at a threshold and its square root makes the logarithms cancel instead: for weakly increasing $F$ with $F(q)\to\infty$ and $q/F(q)\to\infty$, a number $q$ has, on average, exactly $γ$ more divisors below $\sqrt{F(q)}$ than in $[\sqrt{F(q)},F(q))$. For windows on the scale of $q$ itself the average depends on an endpoint convention: with $α\in(0,1)$, counting divisors below $\sqrt{αq}$ against divisors in $(\sqrt{αq},αq)$ yields the average $H_{\lfloor 1/α\rfloor}-\log(1/α)$, while the window $(\sqrt{αq},αq]$ yields $H_{\lceil 1/α\rceil-1}-\log(1/α)$; the two constants differ exactly when $1/α$ takes an integer value, and then by $α$. Either function of $α$ oscillates about $γ$, tends to $γ$ as $α\to0$, and integrates over $(0,1)$ to $ζ(2)-1$. The principal results are formally verified in Lean~4.