# Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions

@inproceedings{Tao2021QuantitativeBF, title={Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions}, author={Terence Tao and Joni Teravainen}, year={2021} }

We establish quantitative bounds on the U[N ] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k, with error terms of shapeO((log logN)−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.

## 3 Citations

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. A surprising ‘converse to the polynomial method’ of Aaronson et al. (CCC’16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a…

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We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial…

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