Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1031-1040

ON ALGEBRAIC DIFFERENTIAL EQUATIONS OF GAMMA FUNCTION AND RIEMANN ZETA FUNCTION

Feng Lü

China University of Petroleum, College of Science
Qingdao, Shandong, 266580, P.R. China; lvfeng18 'at' gmail.com

Abstract. Due to Voronin's universality theorem and Riemann–von Mangoldt formula, this paper concerns the problem of algebraic differential independence between the gamma function Γ and the function f(ζ), where ζ is the Riemann zeta function and f is a function with at least one zero-point. It is showed that Γ and f(ζ) cannot satisfy any nontrivial distinguished differential equation with meromorphic coefficients φ having Nevanlinna characteristic satisfying T(r,φ) = o(r) as r → ∞.

2010 Mathematics Subject Classification: Primary 34M15, 11M06, 33B15, 30D30.

Key words: Voronin's university theorem, the Riemann zeta function, the gamma function, algebraic differential equation.

Reference to this article: F. Lü: On algebraic differential equations of gamma function and Riemann zeta function. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1031-1040.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4455