Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

Will Sawin; Mark Shusterman

doi:10.1007/s00222-022-01115-y

Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

Will Sawin, Mark Shusterman

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

We establish cancellation in short sums of certain special trace functions over F_q[u] below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the F_q[u] -analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in F_q[u].

Original language	English (US)
Pages (from-to)	751-927
Number of pages	177
Journal	Inventiones Mathematicae
Volume	229
Issue number	2
DOIs	https://doi.org/10.1007/s00222-022-01115-y
State	Published - Aug 2022
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00222-022-01115-y

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abstract = "We establish cancellation in short sums of certain special trace functions over Fq[u] below the P{\'o}lya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the Fq[u] -analog of Chowla{\textquoteright}s conjecture on cancellation in M{\"o}bius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in Fq[u].",

author = "Will Sawin and Mark Shusterman",

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AU - Shusterman, Mark

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N2 - We establish cancellation in short sums of certain special trace functions over Fq[u] below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the Fq[u] -analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in Fq[u].

AB - We establish cancellation in short sums of certain special trace functions over Fq[u] below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the Fq[u] -analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in Fq[u].

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Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

Abstract

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