English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We study the problem of obtaining asymptotic formulas for the sums $$\sum _{X < n \le 2X} d_k(n) d_l(n+h)$$∑X<n≤2Xdk(n)dl(n+h) and $$\sum _{X < n \le 2X} \Lambda (n) d_k(n+h)$$∑X<n≤2XΛ(n)dk(n+h), where $$\Lambda $$Λ is the von Mangoldt function, $$d_k$$dk is the $$k^{{\text {th}}}$$kth divisor function, X is large and $$k \ge l \ge 2$$k≥l≥2 are integers. We show that for almost all $$h \in [-H, H]$$h∈[-H,H] with $$H = (\log X)^{10000 k \log k}$$H=(logX)10000klogk, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $$\Lambda (n) \Lambda (n + h)$$Λ(n)Λ(n+h) and we obtained better estimates for the error terms at the price of having to take $$H = X^{8/33 + \varepsilon }$$H=X8/33+ε.

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