An investigation into a family of definite integrals containing log-polylog functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and may be represented as a BBP-type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function.
In this note, we derive the following sum$$ \beta(s) = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}=-\sum_{1\leq i_{1} \leq i_{2} \leq \cdots \leq i_{s}}\frac{2^{i_{s}-1}}{i_{s}\binom{2i_{s}}{i_{s}}}\cdot \frac{1}{(1+2i_{1})(1+2i_{2})\cdots (1+2i_{s})},$$where $s$ is any natural number, and the summation is over all $s$-tuples $(i_{1},i_{2},\cdots,i_{s})$ of natural numbers satisfying $1\leq i_{1}\leq i_{2}\leq \cdots\leq i_{s}$ for fixed $s$.
We derive the following globally convergent series for the Riemann zeta function and the Dirichlet beta function$$\zeta(s)=\frac{1}{2^{s}-2}\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}}\binom{2k+1}{k+1}\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}} \qquad \mbox{(where $s \neq 1+\frac{2\pi i n}{\ln 2}$)},$$$$\beta(s)=\frac{1}{4^{s}}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\left(\frac{3}{4}\right)^{(k+1)}-\left(\frac{1}{4}\right)^{(k+1)}\right)\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}}$$using a globally convergent series for the polylogarithm function, and integrals representing the Riemann zeta function and the Dirichlet beta function. To the best of our knowledge, these series representations are new. Additionally, we give another proof of Hasse's series representation for the Riemann zeta function.
Some BBP-type series for polylog integrals
