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<p>This paper has three purposes. The first one concerns the problem of algebraic independence of Dirichlet series and the Euler gamma function. In addition, by employing Wiman-Valiron theory, we study the problem of algebraic independence of Dirichlet series and some entire functions. Finally, we show that if <i>f</i> is algebraic differentially independent over <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1143_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>, then <i>g</i>(<i>f</i>) is algebraic differentially independent over <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1143_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>, where <i>g</i> is a nonconstant meromorphic function and <i>f</i> is an entire function.</p>

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On algebraic independence of Dirichlet series and the Euler gamma function

  • 期刊论文

  • 发表日期：

    2025年6月23日

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  • Feng Lü,
  • Shuxin Jiang,
  • Weiran Lü

摘要

This paper has three purposes. The first one concerns the problem of algebraic independence of Dirichlet series and the Euler gamma function. In addition, by employing Wiman-Valiron theory, we study the problem of algebraic independence of Dirichlet series and some entire functions. Finally, we show that if f is algebraic differentially independent over \(\mathbb {C}\) C , then g(f) is algebraic differentially independent over \(\mathbb {C}\) C , where g is a nonconstant meromorphic function and f is an entire function.