Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions
Abstract
We establish quantitative bounds on the $U^k[N]$ Gowers norms of the Möbius function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error terms of shape $O((\log\log N)^{c})$. 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 $k$term arithmetic progressions with shifted prime difference.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.02158
 Bibcode:
 2021arXiv210702158T
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Dynamical Systems;
 11B30;
 11N37;
 37A44
 EPrint:
 56 pages