A unified strategy to compute some special functions of number-theoretic interest

We introduce a new algorithm to efficiently compute the functions belonging to a suitable set $scrF$ defined as follows: $fin scrF$ means that $f(s,x)$, $sin Asubset R$ being fixed and $x>0$, has a power series expansion centred at $x_0=1$ with convergence radius greater or equal than $1$; moreover, it satisfies a difference equation of step $1$ and the Euler-Maclaurin summation formula can be applied to $f$. Denoting Euler's function as $Gamma$, we will show, for $x>0$, that $log Gamma(x)$, the digamma function $psi(x)$, the polygamma functions $psi^{(w)}(x)$, $win N$, $wge1$, and, for $s>1$ being fixed, the Hurwitz $zeta(s,x)$-function and its first partial derivative $rac{partialzeta}{partial s}(s,x)$ are in $scrF$. In all these cases the power series involved will depend on the values of $zeta(u)$, $u>1$, where $zeta$ is Riemann's function. As a by-product, we will also show how compute efficiently the Dirichlet $L$-functions $L(s,chi)$ and $L^prime(s,chi)$, $s> 1$, $chi$ being a primitive Dirichlet character, by inserting the reflection formulae of $zeta(s,x)$ and $rac{partialzeta}{partial s}(s,x)$ into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet $eta$-function and for the Catalan constant $G$. Finally, we will study the case of the Bateman $G$-function. In the last section we will also describe some tests that show an important performance gain with respect to a standard multiprecision implementation of $zeta(s,x)$ and $rac{partialzeta}{partial s}(s,x)$, $s>1$, $x>0$.