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- Title
LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER Fq[t].
- Authors
Bienvenu, Pierre‐Yves; Lê, Thái Hoàng
- Abstract
We examine correlations of the Möbius function over Fq[t] with linear or quadratic phases, that is, averages of the form 11qn∑deg f<nμ(f)χ(Q(f)) for an additive character χ over Fq and a polynomial Q∈Fq[x0,…,xn−1] of degree at most 2 in the coefficients x0,…,xn−1 of f=∑i<nxiti. As in the integers, it is reasonable to expect that, due to the random‐like behaviour of 휇, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by O휖(q(−1/4+휖)n) for any 휖>0 if Q is linear and O(q−nc) for some absolute constant c>0 if Q is quadratic. The latter bound may be reduced to O(q−c'n) for some c'>0 when Q(f) is a linear form in the coefficients of f2, that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive‐combinatorial Bogolyubov theorem.
- Publication
Mathematika, 2019, Vol 65, Issue 3, p505
- ISSN
0025-5793
- Publication type
Article
- DOI
10.1112/S0025579319000032