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

TL;DR

This paper introduces a unified algorithmic framework for computing special number-theoretic functions like the gamma, zeta, and L-functions, leveraging power series, functional equations, and FFT techniques for efficiency.

Contribution

The paper develops a novel unified approach to compute various special functions of number theory, including new algorithms for Dirichlet L-functions and constants like Catalan's G, with performance improvements.

Findings

Algorithms outperform standard multiprecision methods

Unified framework applies to multiple special functions

New formulas for Dirichlet beta and Catalan constant

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 $\Gamma$, we will show that, for $x>0$, $\log \Gamma(x)$, the digamma function $\psi(x)$, the polygamma functions $\psi^{(w)}(x)$, $w\in {\mathbb N}$, $w\ge1$, and, for $s>1$ being fixed, the Hurwitz $\zeta(s,x)$-function and its first partial derivative $\frac{\partial\zeta}{\partial s}(s,x)$ are in ${\mathscr F}$. In all these cases the coefficients of the involved power series will depend on the values of $\zeta(u)$, $u>1$, where $\zeta$ is the Riemann zeta-function. As a by-product, we will also show how to compute 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 $\frac{\partial\zeta}{\partial s}(s,x)$ into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet $\beta$-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 $\eta$-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 $\zeta(s,x)$ and $\frac{\partial\zeta}{\partial s}(s,x)$, $s>1$, $x>0$.