Divisor Sums with Spacing Constraints

Abstract

For x,y>0 let S(x,y) denote the largest value of ∑_{n∈ A}τ(n) over subsets A of [1,x]∩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)∼ xlog x, the same asymptotic as the unconstrained sum ∑_{n≤ 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 xlog x uniformly as long as λ≤ 2+2κ/√(loglog 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 ∑_{n≤ 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.

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BibTeX RIS

BibTeX

@misc{chatterjee2026divisor,
  title = {{Divisor Sums with Spacing Constraints}},
  author = {Aahan Chatterjee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31832},
  eprint = {2610.31832},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31832},
}

RIS

TY  - UNPB
TI  - Divisor Sums with Spacing Constraints
AU  - Aahan Chatterjee
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31832
UR  - https://arxiv.org/abs/2610.31832
AB  - For x,y>0 let S(x,y) denote the largest value of ∑_n∈ Aτ(n) over subsets A of [1,x]∩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)∼ xlog x, the same asymptotic as the unconstrained sum ∑_n≤ xτ(n). Near the threshold λ=2 the two regimes meet, and we show that for any fixed κ>0, S(x,y)asymp_κ xlog x uniformly as long as λ≤ 2+2κ/√(loglog 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 ∑_n≤ xτ(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.
ER  -

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