# Quadratic uniformity of the Mobius function

TC Tao, Ben J Green

Research output: Non-textual formWeb publication/site

## Abstract

This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], to establish an asymptotic for the number of four-term progressions p_1 <p_2 <p_3 <p_4 [-1,1] is a Lipschitz function, and if T_g : G/\Gamma -> G/\Gamma is the action of g \in G on G/\Gamma, then the Mobius function \mu(n) is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in g and x in G/\Gamma. This can be viewed as a quadratic'' generalisation of an exponential sum estimate of Davenport, and is proven by the following the methods of Vinogradov and Vaughan.
Translated title of the contribution Quadratic uniformity of the Mobius function English In Preparation Published - 2012

### Bibliographical note

Other: To appear in Annales de l'Institut Fourier