Divisor Sums with Spacing Constraints

Abstract

For $x,y>0$ let $S(x,y)$ denote the largest value of $\sum_{n\in A}τ(n)$ over subsets $A$ of $[1,x]\cap\N$ whose elements are pairwise separated by more than $y$ (here $τ$ is the divisor function). We determine the order of magnitude, and in a subrange the precise asymptotic, of $S(x,y)$ when the spacing parameter $y$ ranges over a polylogarithmic scale $y=(\log x)^{1+λ(\logλ-1)}$, $λ>1$. We show that for every fixed $λ>2$ the spacing constraint costs a computable power of $\log x$ in the exponent, while for every fixed $1<λ<2$ it costs nothing at all: $S(x,y)\sim x\log x$, the same asymptotic as the unconstrained sum $\sum_{n\le x}τ(n)$. Near the threshold $λ=2$ the two regimes meet, and we show that for any fixed $κ>0,$ $S(x,y)\asymp_κ x\log x$ uniformly as long as $λ\le 2+2κ/\sqrt{\log\log x}$. The proof combines the Sathe-Selberg distribution of the number of prime factors, a smooth squarefree estimate obtained from a generating function together with Cauchy's formula, a sieve estimate for correlations among the squarefree parts of the integers that contribute most heavily to $\sum_{n\le x}τ(n)$, and two combinatorial devices (the Caro-Wei bound and a concentration inequality on the symmetric group) used to extract a well-spaced independent set from a sparse auxiliary graph.