×

zbMATH — the first resource for mathematics

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
Citations:
JFM 18.0440.02
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Campbell, R. (1966). Les integrales euleriennes et leur applications, Dunod, Paris (see bibliography on the {\(\Gamma\)}-function). · Zbl 0174.36201
[2] Everitt, W. N. (1944). Some Remarks on the Titchmarsh-Weyl m-coefficient and Associated Differential Operators, Lecture Notes in Pure and Applied Mathematics, Vol. 52 (differential equations, dynamical systems, and control theory) M. Dekker Inc., New York, 33–53. · Zbl 0793.34060
[3] Hölder O. (1886). Über die Eigenschaft der Gammafunktion, keiner algebraischen Differentialgleichung zu genügen. Math. Annal. 28, 1–13 · JFM 18.0440.02
[4] Kolchin, E. Extensions of differential fields I, II, III. Ann Math 43 (1942), 724–729; Ann math. 45 (1944), 358–361; BAMS 53 (1947), 397–401. · Zbl 0060.08105
[5] Kolchin, E. (1973). Differential Algebra and Algebraic Groups, Acad, Press (Extensive bibliography). · Zbl 0264.12102
[6] Markus, L. (2003). Differential Independence of Meromorphic Functions, University of Minnesota Research Report, 1–34.
[7] Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Mon. Nat. Berlin Akad. November, 671–680.
[8] Ritt J.F. (1950). Differential Algebra. AMS Colloq. Publ. NY · Zbl 0037.18402
[9] Titchmarch, E. C. (1951). The Theory of the Riemann Zeta-Function, Oxford Press.
[10] Zariski O., Samuel P. (1958). Commutative Algebra, Vol. I, Ch. II, van Nostrand Co., Princeton NJ · Zbl 0081.26501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.