HYPERTRANSCENDENCE OF -FUNCTIONS FOR
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.