Fast Computing Formulas for some Dirichlet L-Series

TL;DR

This work develops accelerated, provably valid identities for select Dirichlet L-series $L_k(s)$ by expressing them as linear combinations of Hurwitz zeta terms and replacing these with rapidly convergent hypergeometric-type series via a combined Wilf–Zeilberger and Dougall $_5H_5$ approach. The author derives reduction formulas and acceleration seeds for $s=2,3$, producing fast $L_k(s)$ formulas for $k=-7,-8,-15,-20,-24$ at $s=2$ and $k=5,8,12$ at $s=3$, including efficient computations of $\zeta(3)$ and Catalan’s constant $G$. The formulas are numerically validated to $10^8$ digits, and the work provides y-cruncher configuration files to enable practical high-precision assessments on standard hardware. Collectively, these results offer the fastest known hypergeometric-type computations for several Dirichlet L-values and related constants, with broad implications for high-precision analytic number theory.

Abstract

For $χ_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,χ_k)$, expressed as linear combinations of Hurwitz $ζ$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $ζ$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute these selected $L-$functions. The new algorithms include $ζ(3):=L_1(3)$, (Apery$'$s constant), $G:=L_\text{-4}(2)$ (Catalan$'$s constant) as well as $\text{}L_\text{k}(2)\text{}$ for $k=-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k=5, 8, 12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value.