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On the algebraic differential independence of \( \Gamma \) and \( \zeta \)

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Abstract

The paper concerns the problem of algebraic differential independence between Riemann zeta function and functions in a certain class \({\mathcal {F}}\), which contains the gamma function and some exponential functions. It is proved that Riemann zeta function and the functions in \({\mathcal {F}}\) cannot satisfy a class of nontrivial algebraic differential equations.

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References

  1. Bank, S., Kaufman, R.: An extension of Hölder’s theorem concerning the gamma function. Funkc. Ekvacioj 19, 53–63 (1976)

    MATH  Google Scholar 

  2. Cartan, H.: Sur les systémes de fonctions holomorphes á variétés linéaires lacunaires et leurs applications. Ann. Sci. École Norm. Super. 3(45), 255–346 (1928)

    Article  MATH  Google Scholar 

  3. Chen, W., Wang, Q.: On the differential and difference independence of \(\Gamma \) and \(\zeta \). Acta Math. Sci. 41B, 505–516 (2021)

    Article  MathSciNet  Google Scholar 

  4. Chen, W., Wang, Q.: Algebraic differential independence concerning the Euler \(\Gamma \)-function and Dirichlet series. Acta Math. Sci. 40B, 1035–1044 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chiang, Y., Feng, S.: Difference independence of the Riemann zeta function. Acta Arith. 125, 317–329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, Q., Liu, J.: Algebraic differential independence regarding the Euler \(\Gamma \)-function and the Riemann \(\zeta \)-function. J. Number Theory 221, 109–121 (2021)

    Article  MathSciNet  Google Scholar 

  7. Hausdorff, F.: Zum Hölderschen Satzüber \(\zeta (z)\). Math. Ann. 94, 244–247 (1925)

    Article  MathSciNet  Google Scholar 

  8. Hayman, W.: Meromorphic Function. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  9. Hilbert, D.: Mathematische Probleme, Arch. Math. Phys. 1, 44–63, 213–317 (1901)

  10. Hölder, O.: Uber die Eigenschaft der \(\Gamma \)-function, keiner algebraischen Differentialgleichung zu genügen. Math. Ann. 28, 1–13 (1887)

    Article  Google Scholar 

  11. Huang, J., Ng, T.: Hypertranscendency of perturbations of hypertranscendental functions. J. Math. Anal. Appl. 491, 124390 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levin, B.: Distribution of zeros of entire functions, rev. ed., transl. from the Russian by Boas R.P. et al. Am. Math. Soc. Providence, (1980)

  13. Li, B., Ye, Z.: Algebraic differential equations concerning the Riemann zeta function and the Euler gamma function. Indiana Univ. Math. J. 59, 1405–1415 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, B., Ye, Z.: On algebraic differential properties of the Riemann \(\zeta \)-function and Euler \(\Gamma \)-function. Complex Var. Elliptic Equ. 56, 137–145 (2011)

    Article  MathSciNet  Google Scholar 

  15. Li, B., Ye, Z.: Algebraic differential equations with functional coefficients concerning \(\zeta \) and \(\Gamma \). J. Differ. Equ. 260, 1456–1464 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liao, L., Yang, C.: On some new properties of the gamma function and the Riemann zeta-function. Math. Nachr. 257, 59–66 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lü, F.: On algebraic differential equations for the gamma function and \(L\)-functions in the extended Selberg class. Bull. Aust. Math. Soc. 96, 36–43 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lü, F.: A study on algebraic differential equationd of gamma function and Dirichlet series. J. Math. Anal. Appl. 462, 1195–1204 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lü, F.: On algebraic differential equations of gamma function and Riemann zeta function. Ann. Acad. Sci. Fenn. Math. 44, 1031–1040 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Markus, L.: Differential independence of \(\Gamma \) and \(\zeta \). J. Dyn. Differ. Equ. 19, 133–154 (2007)

    Article  MathSciNet  Google Scholar 

  21. Matsumoto, K.: A survey on the theory of universality for zeta and \(L\)-functions. In: Number Theory, 95–144, Ser. Number Theory Appl. 11. World Science Publications, Hackensack, NJ (2015)

  22. Mordykhai-Boltovskoi, D.: On hypertranscendence of the function \(\xi (x, s)\). Izv. Politekh. Inst. Warsaw 2, 1–16 (1914)

    Google Scholar 

  23. Moore, E.: Concerning transcendentally transcendental functions. Math. Ann. 48, 49–74 (1897)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ostrowski, A.: Neuer Beweis des Hölderschen Satzes, da\(\beta \) die Gammafunktion keiner algebraischen Differffentialgleichung genügt. Math. Ann. 79, 286–288 (1918)

    Article  MathSciNet  Google Scholar 

  25. Ostrowski, A.: Über Dirichletsche Reihen und algebraische Differentialgleichungen. Math. Z. 8, 241–298 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ostrowski, A.: Zum Hölderschen Satzüber \(\Gamma (z)\). Math. Ann. 94, 248–251 (1925)

    Article  MathSciNet  Google Scholar 

  27. Steuding, J.: Value Distribution of \(L\)-Functions. Lecture Notes in Math, vol. 1877. Springer, Berlin (2007)

  28. Titchmarsh, E.: The Theory of Functions (2ND edN). Oxford University Press (1968), reprinted

  29. Voronin, S.: The distribution of the nonzero values of the Riemann \(\zeta \)-function. Trudy Mat. Inst. Steklov. 128, 131–150 (1972)

    MathSciNet  Google Scholar 

  30. Voronin, S.: A theorem on the distribution of values of the Riemann \(\zeta \)-function. Dokl. Akad. Nauk SSSR. 221, 771 (1975)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for helpful suggestions.

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Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266580, Shandong, People’s Republic of China

    Feng Lü & Shuxin Jiang

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  1. Feng Lü
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  2. Shuxin Jiang
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Correspondence to Feng Lü.

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Communicated by Alberto Minguez.

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Research supported by Natural Science Foundation of Shandong Province (ZR2022MA014)

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Lü, F., Jiang, S. On the algebraic differential independence of \( \Gamma \) and \( \zeta \). Monatsh Math 202, 141–159 (2023). https://doi.org/10.1007/s00605-023-01858-7

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