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

Abstract

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ 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 functional equation of step $1$ and the Euler-Maclaurin summation formula can be applied to $f$. Denoting the Euler gamma-function as $Γ$, we will show that, for $x>0$, $\log Γ(x)$, the digamma function $ψ(x)$, the polygamma functions $ψ^{(w)}(x)$, $w\in {\mathbb N}$, $w\ge1$, and, for $s>1$ being fixed, the Hurwitz $ζ(s,x)$-function and its first partial derivative $\frac{\partialζ}{\partial s}(s,x)$ are in ${\mathscr F}$. In all these cases the coefficients of the involved power series will depend on the values of $ζ(u)$, $u>1$, where $ζ$ is the Riemann zeta-function. As a by-product, we will also show how to compute the Dirichlet $L$-functions $L(s,χ)$ and $L^\prime(s,χ)$, $s> 1$, $χ$ being a primitive Dirichlet character, by inserting the reflection formulae of $ζ(s,x)$ and $\frac{\partialζ}{\partial s}(s,x)$ into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet $β$-function and for the Catalan constant $G$. Finally, we will study the case of the Bateman $G$-function and of the alternating Hurwitz zeta-function, also known as the $η$-function; we will show that, even if they are not in ${\mathscr F}$, our approach can be adapted to handle them too. In the last section we will also describe some tests that show a performance gain with respect to a standard multiprecision implementation of $ζ(s,x)$ and $\frac{\partialζ}{\partial s}(s,x)$, $s>1$, $x>0$.