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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.

##### 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
JFM 18.0440.02
Full Text:
##### References:
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