Beyond the Erdős discrepancy problem in function fields

Abstract

Abstract We characterize the limiting behavior of partial sums of multiplicative functions $$f:\mathbb {F}_q[t]\rightarrow S^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> . In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals , short intervals , or lexicographic intervals . Concerning the notion of short interval discrepancy, we show that a completely multiplicative $$f:\mathbb {F}_q[t]\rightarrow \{-1,+1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with q odd has bounded short interval sums if and only if f coincides with a “modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> that such modified characters are extremal with respect to partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of $$\mathbb {F}_{q}[t]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on $$\mathbb {F}_q[t]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.

Locations

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