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$L$-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.
This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations 
$af(z^{r})=f(z)+R(z)$, where 
$a$ is a nonzero complex number, 
$r$ an integer greater than 1, and 
$R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.
