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LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER F q [ t ]
Author(s) -
Bienvenu PierreYves,
Lê Thái Hoàng
Publication year - 2019
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579319000032
Subject(s) - mathematics , bounded function , quadratic function , quadratic equation , constant (computer programming) , function (biology) , quadratic form (statistics) , combinatorics , bilinear interpolation , polynomial , pure mathematics , discrete mathematics , mathematical analysis , statistics , geometry , evolutionary biology , computer science , biology , programming language
We examine correlations of the Möbius function overF q [ t ]with linear or quadratic phases, that is, averages of the form 11 q n∑ deg f < nμ ( f ) χ ( Q ( f ) )for an additive character χ over F q and a polynomial Q ∈ F q [ x 0 , … , x n − 1 ]of degree at most 2 in the coefficientsx 0 , … , x n − 1of f = ∑ i < nx i t i . 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 byO ( q ( − 1 / 4 + ) n )for any > 0 if Q is linear and O ( q − n c) for some absolute constant c > 0 if Q is quadratic. The latter bound may be reduced to O ( q − c ' n ) for somec ' > 0 when Q ( f ) is a linear form in the coefficients of f 2 , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive‐combinatorial Bogolyubov theorem.