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

Abstract

We study the problem of obtaining asymptotic formulas for the sums $$\sum _{X < n \le 2X} d_k(n) d_l(n+h)$$ and $$\sum _{X < n \le 2X} \Lambda (n) d_k(n+h)$$ , where $$\Lambda $$ is the von Mangoldt function, $$d_k$$ is the $$k^{{\text {th}}}$$ divisor function, X is large and $$k \ge l \ge 2$$ are integers. We show that for almost all $$h \in [-H, H]$$ with $$H = (\log X)^{10000 k \log k}$$ , the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $$\Lambda (n) \Lambda (n + h)$$ and we obtained better estimates for the error terms at the price of having to take $$H = X^{8/33 + \varepsilon }$$ .

Locations

• Mathematische Annalen - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF
• arXiv (Cornell University) - View - PDF
• Mathematische Annalen - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF
• arXiv (Cornell University) - View - PDF
• Mathematische Annalen - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF
• arXiv (Cornell University) - View - PDF