LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER Fq[t]

We examine correlations of the Möbius function over Fq[t] with linear or quadratic phases, that is, averages of the form 11qn∑deg f0 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 quadr...

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Veröffentlicht in:	Mathematika 2019, Vol.65 (3), p.505-529
Hauptverfasser:	Bienvenu, Pierre‐Yves, Lê, Thái Hoàng
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Sprache:	eng
Schlagworte:	11B30 (primary) 11T55 (secondary)
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creator	Bienvenu, Pierre‐Yves Lê, Thái Hoàng
description	We examine correlations of the Möbius function over Fq[t] with linear or quadratic phases, that is, averages of the form 11qn∑deg f0 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.
doi_str_mv	10.1112/S0025579319000032
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As in the integers, it is reasonable to expect that, due to the random‐like behaviour of μ, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by Oε(q(−1/4+ε)n) for any ε>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.</description><identifier>ISSN: 0025-5793</identifier><identifier>EISSN: 2041-7942</identifier><identifier>DOI: 10.1112/S0025579319000032</identifier><language>eng</language><publisher>London, UK: London Mathematical Society</publisher><subject>11B30 (primary) ; 11T55 (secondary)</subject><ispartof>Mathematika, 2019, Vol.65 (3), p.505-529</ispartof><rights>2019 University College London</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1112%2FS0025579319000032$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1112%2FS0025579319000032$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>314,780,784,1417,4024,27923,27924,27925,45574,45575</link.rule.ids></links><search><creatorcontrib>Bienvenu, Pierre‐Yves</creatorcontrib><creatorcontrib>Lê, Thái Hoàng</creatorcontrib><title>LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER Fq[t]</title><title>Mathematika</title><description>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 μ, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by Oε(q(−1/4+ε)n) for any ε>0 if Q is linear and O(q−nc) for some absolute constant c>0 if Q is quadratic. 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title	LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER Fq[t]
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