TY - JOUR

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

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

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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U2 - 10.1007/s00208-018-01801-4

DO - 10.1007/s00208-018-01801-4

M3 - Article

AN - SCOPUS:85063197593

SN - 0025-5831

VL - 374

SP - 793

EP - 840

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -