Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory pp 183-193 | Cite as

# 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 \(q=e^{-\pi }\), \(e^{-2\pi }\).

## Notes

### Acknowledgements

We thank Krishnaswami Alladi and Ankush Goswami for their beautiful ideas and contributions. We also thank Zhi-Wei Sun for inspiring this work.

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