Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges

Kaisa Matomäki, Maksym Radziwiłł*, Terence Tao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

We study the problem of obtaining asymptotic formulas for the sums ∑X<n≤2X dk (n)dl (n + h) and ∑X<n≤2X⋀(n)dk(n + h), where ⋀ is the von Mangoldt function, dk is the kth divisor function, X is large and k ≥ l ≥ 2 are integers. We show that for almost all h ∈[−H, H] with H = (log X)10000klogk, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of ⋀(n)⋀ (n + h) and we obtained better estimates for the error terms at the price of having to take H = X8/33+ε.

Original languageEnglish (US)
Pages (from-to)793-840
Number of pages48
JournalMathematische Annalen
Volume374
Issue number1-2
DOIs
StatePublished - Jun 2019

ASJC Scopus subject areas

  • General Mathematics

Fingerprint

Dive into the research topics of 'Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges'. Together they form a unique fingerprint.

Cite this