Sums of infinite series involving the Dirichlet lambda function

Abstract

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In this paper, by applying the partial fraction decomposition of $\pi \tan(\pi x)$ and explicit evaluations of the integrals \[ \int_0^{\frac{1}{2}} x^{2m-1} \cos(2l\pi x) dx \quad \text{and} \quad \int_0^{\frac{1}{2}} x^{m-1} \log \cos(\pi x) dx, \] for positive integers $l$ and $m$, we derive closed-form expressions for several classes of infinite series involving $\lambda(s)$. We also demonstrate that the values $\lambda(k)$ for even integers $k \geq 2$ arise as constant terms in the Fourier expansions of Eisenstein series associated with the congruence subgroup \[ \Gamma_0(2) := \left\{ \begin{pmatrix} a & b c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) : c \equiv 0 \pmod{2} \right\}. \]