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HYPERTRANSCENDENCE OF $L$ -FUNCTIONS FOR $\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$


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.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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