Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

  • Terence Tao

    University of California, Los Angeles, USA
  • Joni Teräväinen

    University of Turku, Turku, Finland
Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions cover

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Abstract

We establish quantitative bounds on the Gowers norms of the Möbius function  and the von Mangoldt function for all , with error terms of the shape . As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no -term arithmetic progressions with shifted prime difference.

Cite this article

Terence Tao, Joni Teräväinen, Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1404