Local divisor correlations in almost all short intervals

2025/03/04 by Pliego, Javier, Sun, Yu-Chen, Wang, Mengdi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.02962

Abstract

Let k,l ≥ 2 be natural numbers, and let d_k,d_l denote the k-fold and l-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum ∑_x0 be a small fixed number and let Φ(x) be a positive function that tends to infinity arbitrarily slowly as x→ ∞. We then show that whenever H₁≥(log x)^Φ(x) and (log x)^{1000klog k}≤ H₂≤ H₁^1-ε , the expected asymptotic formula holds for almost all x∈[X,2X] and almost all 1≤ h≤ H₂.

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