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HYPERTRANSCENDENCE OF
$L$-FUNCTIONS FOR 
$\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 11 November 2015, pp. 388-399
- Print publication:
- June 2016
-
- Article
-
- You have access
- Export citation
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We generalise a result of Hilbert which asserts that the Riemann zeta-function
${\it\zeta}(s)$ is hypertranscendental over 
$\mathbb{C}(s)$. Let 
${\it\pi}$ be any irreducible cuspidal automorphic representation of 
$\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We establish a certain type of functional difference–differential independence for the associated 
$L$-function 
$L(s,{\it\pi})$. This result implies algebraic difference–differential independence of 
$L(s,{\it\pi})$ over 
$\mathbb{C}(s)$ (and more strongly, over a certain field 
${\mathcal{F}}_{s}$ which contains 
$\mathbb{C}(s)$). In particular, 
$L(s,{\it\pi})$ is hypertranscendental over 
$\mathbb{C}(s)$. We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.
