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 $c\log B+(\gamma -c)+O(B^{-\min (c,1-c)})$, where $\gamma$ 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 $\gamma$ 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 $\gamma$. 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 $\gamma$ 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 $\alpha \in (0,1)$, counting divisors below $\sqrt{\alpha q}$ against divisors in $(\sqrt{\alpha q},\alpha q)$ yields the average $H_{\lfloor 1/\alpha \rfloor }-\log (1/\alpha )$, while the window $(\sqrt{\alpha q},\alpha q]$ yields $H_{\lceil 1/\alpha \rceil -1}-\log (1/\alpha )$; the two constants differ exactly when $1/\alpha$ takes an integer value, and then by $\alpha$. Either function of $\alpha$ oscillates about $\gamma$, tends to $\gamma$ as $\alpha \to 0$, and integrates over $(0,1)$ to $\zeta (2)-1$. The principal results are formally verified in Lean~4.

10 pages