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<p>We establish the first case of the surprising correlation phenomenon observed in the recent works of He, Lee, Oliver, Pozdnyakov, and Sutherland between Fourier coefficients in families of modular forms and their root numbers. We give a complete description of the resulting correlation functions for holomorphic modular forms of any fixed weight <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1347_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> <EquationSource Format="TEX">$k$</EquationSource> </InlineEquation> and examine the asymptotic properties of these functions.</p>

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Murmurations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月16日

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  • Nina Zubrilina

摘要

We establish the first case of the surprising correlation phenomenon observed in the recent works of He, Lee, Oliver, Pozdnyakov, and Sutherland between Fourier coefficients in families of modular forms and their root numbers. We give a complete description of the resulting correlation functions for holomorphic modular forms of any fixed weight k $k$ and examine the asymptotic properties of these functions.