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<p>We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1343_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$(\log X)^{\psi (X)}$</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1343_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$\psi (X) \rightarrow \infty $</EquationSource> </InlineEquation> an arbitrarily slowly growing function of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1343_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> <EquationSource Format="TEX">$X$</EquationSource> </InlineEquation>. We expect the methods should adapt to nilsequences, thus also showing that the Generalised Riemann Hypothesis implies close to exponential growth in the sign patterns of the Liouville function.</p>

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Stability under scaling in the local phases of multiplicative functions

  • 期刊论文

  • 发表日期：

    2025年6月16日

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  • Miguel N. Walsh

摘要

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least ( log X ) ψ ( X ) $(\log X)^{\psi (X)}$ , with ψ ( X ) → ∞ $\psi (X) \rightarrow \infty $ an arbitrarily slowly growing function of X $X$ . We expect the methods should adapt to nilsequences, thus also showing that the Generalised Riemann Hypothesis implies close to exponential growth in the sign patterns of the Liouville function.