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<p>In this article, we investigate the behaviour of the values of the zeta sums <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_658_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sum _{n\le x}n^{\textrm{i}t}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msup> <mi>n</mi> <mrow> <mtext>i</mtext> <mi>t</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> if <i>x</i> and <i>t</i> are large. We show some asymptotic behaviour and Omega results on zeta sums that are analogous to previous results on the large character sums <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_658_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sum _{n\le x}\chi (n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Large zeta sums

  • 期刊论文

  • 发表日期：

    2025年6月10日

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  • Zikang Dong,
  • Weijia Wang,
  • Hao Zhang

摘要

In this article, we investigate the behaviour of the values of the zeta sums \(\sum _{n\le x}n^{\textrm{i}t}\) ∑ n ≤ x n i t if x and t are large. We show some asymptotic behaviour and Omega results on zeta sums that are analogous to previous results on the large character sums \(\sum _{n\le x}\chi (n)\) ∑ n ≤ x χ ( n ) .