<p>In this note, we prove a conjecture of Xu on alternating double zeta values by using Glanois’s motive theorem. Namely, we will prove that the linear combination <Equation ID="Equ3"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1759_Article_Equ3.gif" Format="GIF" Height="47" Rendition="HTML" Resolution="72" Type="Linedraw" Width="430" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \zeta (2,\overline{2k})+2k\zeta (1,\overline{2k+1}):=\sum _{0<n<m} \frac{(-1)^{m}}{n^2m^{2k}}+2k\sum _{0<n<m} \frac{(-1)^{m}}{nm^{2k+1}} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mover> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mover> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mn>0</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mi>m</mi> </mrow> </munder> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <msup> <mi>m</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>k</mi> <munder> <mo>∑</mo> <mrow> <mn>0</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mi>m</mi> </mrow> </munder> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mrow> <mi>n</mi> <msup> <mi>m</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>can be expressed in terms of multiple zeta values of depth <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1759_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Proof of Xu conjecture on alternating double zeta values

期刊论文
发表日期：

2025年7月10日

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Wenxuan Zheng,
Ying Yang

摘要

In this note, we prove a conjecture of Xu on alternating double zeta values by using Glanois’s motive theorem. Namely, we will prove that the linear combination $\begin{aligned} \zeta (2,\overline{2k})+2k\zeta (1,\overline{2k+1}):=\sum _{0<n<m} \frac{(-1)^{m}}{n^2m^{2k}}+2k\sum _{0<n<m} \frac{(-1)^{m}}{nm^{2k+1}} \end{aligned}$ $\begin{matrix} ζ (2, \bar{2 k}) + 2 k ζ (1, \bar{2 k + 1}) : = \sum_{0 < n < m} \frac{{(- 1)}^{m}}{n^{2} m^{2 k}} + 2 k \sum_{0 < n < m} \frac{{(- 1)}^{m}}{n m^{2 k + 1}} \end{matrix}$ can be expressed in terms of multiple zeta values of depth $\le 2$ $\leq 2$ .