This analysis demonstrates geometric convergence of infinite series involving quadratic irrationals, highlighting connections with Dirichlet L-functions.
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·)):=∑ₖ₌₁^∞( d/k )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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Sun et al. (2025) studied this question.
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