Higher uniformity of arithmetic functions in short intervals I. All intervals

Kaisa Matomäki; Xuancheng Shao; Terence Tao; Joni Teräväinen

doi:10.1017/fmp.2023.28

Higher uniformity of arithmetic functions in short intervals I. All intervals

Kaisa Matomäki
, Xuancheng Shao
, Terence Tao
, Joni Teräväinen

Mathematics

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

5 Scopus citations

Abstract

We study higher uniformity properties of the Möbius function, the von Mangoldt function, and the divisor functions on short intervals with for a fixed constant. More precisely, letting and any 0$ and be suitable approximants of and and, we show for instance that, for any nilsequence, we have <![CDATA[begin{align}sum{X when and or and. As a consequence, we show that the short interval Gowers norms are also asymptotically small for any fixed s for these choices of. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type sums.

Original language	English
Article number	e29
Journal	Forum of Mathematics, Pi
Volume	11
DOIs	https://doi.org/10.1017/fmp.2023.28
State	Published - Oct 19 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press.

Funding

KM was supported by Academy of Finland grant no. 285894. XS was supported by NSF grant DMS-1802224. 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-1764034. JT was supported by a Titchmarsh Fellowship, Academy of Finland grant no. 340098, and funding from European Union\u2019s Horizon Europe research and innovation programme under Marie Sk\u0142odowska-Curie grant agreement No 101058904.

Funders	Funder number
James and Carol Collins Chair	DMS-1764034, 340098
European Commission	101058904
European Union’s Horizon Europe research and innovation programme	101058904
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1764034, DMS-1802224
Academy of Finland	285894

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1017/fmp.2023.28

Cite this

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abstract = "We study higher uniformity properties of the M{\"o}bius function, the von Mangoldt function, and the divisor functions on short intervals with for a fixed constant. More precisely, letting and any 0\$ and be suitable approximants of and and, we show for instance that, for any nilsequence, we have begin\{align\}sum\{X when and or and. As a consequence, we show that the short interval Gowers norms are also asymptotically small for any fixed s for these choices of. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type sums.</p>",

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T1 - Higher uniformity of arithmetic functions in short intervals I. All intervals

AU - Matomäki, Kaisa

AU - Shao, Xuancheng

AU - Tao, Terence

AU - Teräväinen, Joni

PY - 2023/10/19

Y1 - 2023/10/19

N2 - We study higher uniformity properties of the Möbius function, the von Mangoldt function, and the divisor functions on short intervals with for a fixed constant. More precisely, letting and any 0$ and be suitable approximants of and and, we show for instance that, for any nilsequence, we have begin{align}sum{X when and or and. As a consequence, we show that the short interval Gowers norms are also asymptotically small for any fixed s for these choices of. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type sums.</p>

AB - We study higher uniformity properties of the Möbius function, the von Mangoldt function, and the divisor functions on short intervals with for a fixed constant. More precisely, letting and any 0$ and be suitable approximants of and and, we show for instance that, for any nilsequence, we have begin{align}sum{X when and or and. As a consequence, we show that the short interval Gowers norms are also asymptotically small for any fixed s for these choices of. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type sums.</p>

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JO - Forum of Mathematics, Pi

JF - Forum of Mathematics, Pi

M1 - e29

ER -

Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract

Bibliographical note

Funding

ASJC Scopus subject areas

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