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

Markus, Lawrence

J. Dyn. Differ. Equations 19, No. 1, 133-154 (2007).

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.

Reviewer: Roelof Koekoek (Delft)

MSC:

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

Keywords:

Gamma function; Riemann zeta function; differential algebra

Citations:

JFM 18.0440.02

References:

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