LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

Abstract

We examine correlations of the Möbius function over with linear or quadratic phases, that is, averages of the form (1) for an additive character χ over and a polynomial of degree at most 2 in the coefficients of . As in the integers, it is reasonable to expect that, due to the random-like behaviour of , such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by for any if Q is linear and for some absolute constant if Q is quadratic. The latter bound may be reduced to for some when is a linear form in the coefficients of , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View
• Mathematika - View - PDF