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ON ALGEBRAIC DIFFERENTIAL EQUATIONS FOR THE GAMMA FUNCTION AND $L$-FUNCTIONS IN THE EXTENDED SELBERG CLASS

Part of: Elementary classical functions Entire and meromorphic functions, and related topics Zeta and $L$-functions: analytic theory Differential equations in the complex domain

Published online by Cambridge University Press:  13 March 2017

FENG LÜ
FENG LÜ*
Affiliation:
College of Science, China University of Petroleum, Qingdao, Shandong, 266580, PR China email lvfeng18@gmail.com
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Abstract

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This paper concerns the problem of algebraic differential independence of the gamma function and ${\mathcal{L}}$-functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the ${\mathcal{L}}$-function in a domain $D:=\{z:0<\text{Re}\,z<\unicode[STIX]{x1D70E}_{0}\}$ for a positive constant $\unicode[STIX]{x1D70E}_{0}$.

Keywords

L-functionRiemann zeta functiongamma functionalgebraic differential independence

MSC classification

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 30D30: Meromorphic functions, general theory 33B15: Gamma, beta and polygamma functions 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 36 - 43
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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