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October 2, 2025Open Access

Fast converging irrational series for L (2, (d) )

Key Points

The study reveals fast converging series for several Dirichlet L-functions, including L_{-56}(2) and L_{-68}(2), enhancing existing theories.
Applying Kronecker's theorem leads to new convergent series beyond previous Guillera-Rogers results, revealing deeper connections with quadratic irrationals.
The evaluation utilizes principles from the theory of Guillera-Rogers to establish relationships with special values of L-functions.
Geometric convergence of the series is achieved, showing important advancements in the analysis of irrational series related to number theory.

Abstract

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of π and special values of Dirichlet L-functions Ld (2) L (2, (d) ): =₊=₁^ (dk) 1k². Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for L-₅₆ (2), L-₆₈ (2), L-₈₇ (2), L-₁₁₁ (2), and L-₁₁₆ (2), which go beyond the solvable cases of Guillera-Rogers.

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