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

Terence Tao, Joni Teräväinen

Published 2021-07-05Version 1

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius 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.