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On some new properties of the gamma function and the Riemann zeta function
Author(s) -
Liao Liangwen,
Yang ChungChung
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310078
Subject(s) - mathematics , riemann zeta function , riemann hypothesis , prime zeta function , arithmetic zeta function , functional equation , particular values of riemann zeta function , function (biology) , riemann xi function , pure mathematics , prime (order theory) , gamma function , mathematical analysis , differential equation , combinatorics , evolutionary biology , biology
In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Γ( z ) and the Riemann zeta function ζ( z ). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Γ( z ), does not satisfy any algebraic differential equation with coefficients in ℒ 0 . Moreover, the two functions do not satisfy any functional equation of the form P (Γ, ζ, z ) ≡ 0, where P ( x , y , z ) is a nonconstant polynomial in x , y and z .