CM Evaluations of the Goswami-Sun Series

Abstract

In recent work, Sun constructed two q-series, and he showed that their limits as $q\rightarrow 1$ 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 $\varGamma \left( \frac{1}{2}\right) =\sqrt{\pi }$, it is natural to seek further specializations of these series which involve special values of the $\varGamma$-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 $\varGamma$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $\varGamma \left( \frac{1}{4}\right) ^4/\pi ^3$ when $q=e^{-\pi }$, $e^{-2\pi }$.