Differential independence of \(\Gamma\) and \(\zeta\). (English) Zbl 1119.33004

In [O. Hölder, Über die Eigenshaft der Gammafunktion keiner algebraischen Differentialgleichung zu genügen, Math. Annal. 28, 1–13 (1886; JFM 18.0440.02)] it was shown that Euler’s Gamma function is differential transcendental, id est \(\Gamma(z)\) satisfies no (non-trivial) algebraic ordinary differential equation with coefficients which are polynomials in \(z\). The author tries to extend Hölder’s methods to the Riemann zeta function, which leads to the conjecture that \(\Gamma(z)\) and \(\zeta(z)\) are differential independent, which means that \(\Gamma(z)\) is not a solution of an algebraic differential equation with coefficients that are differential polynomials in \(\zeta(z)\). Although the author is not able to prove this, he shows the partial result that \(\Gamma(z)\) and \(\zeta(\sin 2\pi z)\) are differential independent.


33B15 Gamma, beta and polygamma functions
33E20 Other functions defined by series and integrals
12H05 Differential algebra
13N99 Differential algebra


JFM 18.0440.02
Full Text: DOI


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