CM Evaluations of the Goswami-Sun Series

Abstract

In recent work, Sun constructed two $q$-series, and he showed that their limits as $q\rightarrow1$ give new derivations of the Riemann-zeta values $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. Goswami extended these series to an infinite family of $q$-series, which he analogously used to obtain new derivations of the evaluations of $\zeta(2k)\in\mathbb{Q}\cdot\pi^{2k}$ for every positive integer $k$. Since it is well known that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$, it is natural to seek further specializations of these series which involve special values of the $\Gamma$-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points $\tau$, where $q:=e^{2\pi i\tau}$, are algebraic multiples of specific ratios of $\Gamma$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $\Gamma\left(\frac{1}{4}\right)^4/\pi^3$ when $q=e^{-\pi}$, $e^{-2\pi}$.