Abstract <p>In the paper, it is investigated whether for real argument <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(s\)</EquationSource>  </InlineEquation> the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\((s-1)^{n+1}\)</EquationSource>  </InlineEquation> weighted Riemann zeta function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\zeta^{(n)}(s)\)</EquationSource>  </InlineEquation> limits <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(s\downarrow 1\)</EquationSource>  </InlineEquation> do exist. Here we will look only at <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=0,1\)</EquationSource>  </InlineEquation>. The answer to the question could very well be that assuming existence of limits to be true gives a conflicting outcome. This result may support the possibility of what can be called incompleteness in concrete mathematics. Interestingly, the uncovered incompleteness occurs in an approximation-theoretical framework. The key argument we employ in the paper revolves around the approximation of a l’Hôpital limit. Only an ‘‘after the fact’’ alternative approach reorders the formulae in the derivation. This reordering is, in fact, based on a <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(0=1\)</EquationSource>  </InlineEquation> contradiction as starting point. However, the alternative <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8235_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(0=1\)</EquationSource>  </InlineEquation> based approach is not the limit approximation that is used in the flow of the derivation in the present paper.</p>

Weighted Riemann Zeta Function Limits on the Real Axis

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发表日期：

2025年3月1日

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J. F. Geurdes

摘要

Abstract

In the paper, it is investigated whether for real argument \(s\) the \((s-1)^{n+1}\) weighted Riemann zeta function \(\zeta^{(n)}(s)\) limits \(s\downarrow 1\) do exist. Here we will look only at \(n=0,1\) . The answer to the question could very well be that assuming existence of limits to be true gives a conflicting outcome. This result may support the possibility of what can be called incompleteness in concrete mathematics. Interestingly, the uncovered incompleteness occurs in an approximation-theoretical framework. The key argument we employ in the paper revolves around the approximation of a l’Hôpital limit. Only an ‘‘after the fact’’ alternative approach reorders the formulae in the derivation. This reordering is, in fact, based on a \(0=1\) contradiction as starting point. However, the alternative \(0=1\) based approach is not the limit approximation that is used in the flow of the derivation in the present paper.