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ON ALGEBRAIC DIFFERENTIAL EQUATIONS FOR THE GAMMA FUNCTION AND
$L$-FUNCTIONS IN THE EXTENDED SELBERG CLASS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 13 March 2017, pp. 36-43
- Print publication:
- August 2017
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- Article
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- You have access
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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 
$\unicode[STIX]{x1D70E}_{0}$.
