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The a-Points of the Riemann Zeta-Function and the Functional Equation

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We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the a-points of the zeta-function, i.e., the roots of the equation \(\zeta (s)=a\), where a is an arbitrary fixed complex number.


  • Riemann zeta-function
  • Riemann hypothesis
  • a-points
  • Functional equation

Dedicated to the memory of Professor Eduard Wirsing

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  1. 1.

    The authors’ translation of the German original text: “Es sind bei einer analytischen Funktion die Punkte, an denen sie 0 ist, zwar sehr wichtig; ebenso interessant sind aber die Punkte, an denen sie einen bestimmten Wertaannimmt.


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The first named author was supported by FWF project M 3246-N. The third author was supported by JSPS KAKENHI Grant Numbers 18K13400 and 22K13895, and also MEXT Initiative for Realizing Diversity in the Research Environment.

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Correspondence to Ade Irma Suriajaya .

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Sourmelidis, A., Steuding, J., Suriajaya, A.I. (2023). The a-Points of the Riemann Zeta-Function and the Functional Equation. In: Maier, H., Steuding, J., Steuding, R. (eds) Number Theory in Memory of Eduard Wirsing. Springer, Cham.

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