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

Kaisa Matomäki; Maksym Radziwiłł; Terence Tao

doi:10.1007/s00208-018-01801-4

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

Mathematics

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

We study the problem of obtaining asymptotic formulas for the sums ∑_X<n≤2X d_k (n)d_l (n + h) and ∑_X<n≤2X_⋀(n)d_k(n + h), where ⋀ is the von Mangoldt function, d_k is the k^th 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 = X^8/33+ε.

Original language	English (US)
Pages (from-to)	793-840
Number of pages	48
Journal	Mathematische Annalen
Volume	374
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-018-01801-4
State	Published - Jun 2019

Funding

KM was supported by Academy of Finland Grant no. 285894. MR was supported by a NSERC DG grant, the CRC program and a Sloan Fellowship. TT was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF Grant DMS-1266164. Part of this paper was written while the authors were in residence at MSRI in Spring 2017, which is supported by NSF Grant DMS-1440140. Acknowledgements KM was supported by Academy of Finland Grant no. 285894. MR was supported by a NSERC DG grant, the CRC program and a Sloan Fellowship. TT was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF Grant DMS-1266164. Part of this paper was written while the authors were in residence at MSRI in Spring 2017, which is supported by NSF Grant DMS-1440140.

ASJC Scopus subject areas

General Mathematics

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title = "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 ∑X dk (n)dl (n + h) and ∑X⋀(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.",

author = "Kaisa Matom{\"a}ki and Maksym Radziwi{\l}{\l} and Terence Tao",

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T1 - Correlations of the von Mangoldt and higher divisor functions II

T2 - divisor correlations in short ranges

AU - Matomäki, Kaisa

AU - Radziwiłł, Maksym

AU - Tao, Terence

PY - 2019/6

Y1 - 2019/6

N2 - We study the problem of obtaining asymptotic formulas for the sums ∑X dk (n)dl (n + h) and ∑X⋀(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.

AB - We study the problem of obtaining asymptotic formulas for the sums ∑X dk (n)dl (n + h) and ∑X⋀(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.

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

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