<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_668_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$K/\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </math></EquationSource> </InlineEquation> be a Galois extension, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_668_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$a_K(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the Dirichlet coefficients of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_668_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta _K(s)/\zeta (s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over number field. We study twisted sum of isotypic trace functions associated with some sheaves modulo prime <i>q</i> of bounded conductor with the coefficients of Dedekind zeta function. We are able to establish a non-trivial bound for the algebraic twisted sum in short intervals.</p>

Cancellation in algebraic twisted sums over number field

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发表日期：

2025年7月28日

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Yuxuan Zhou

摘要

Let $K/\mathbb {Q}$ $K / Q$ be a Galois extension, and $a_K(n)$ $a_{K} (n)$ be the Dirichlet coefficients of $\zeta _K(s)/\zeta (s)$ $ζ_{K} (s) / ζ (s)$ over number field. We study twisted sum of isotypic trace functions associated with some sheaves modulo prime q of bounded conductor with the coefficients of Dedekind zeta function. We are able to establish a non-trivial bound for the algebraic twisted sum in short intervals.