Quadratic uniformity of the Mobius function

TC Tao, Ben J Green

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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 contributionQuadratic uniformity of the Mobius function
Original languageEnglish
EditionIn Preparation
Publication statusPublished - 2012

Bibliographical note

Other: To appear in Annales de l'Institut Fourier


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