A Twist of a Ramanujan Identity

15 Jan 2020 Chu Hung Viet Chu Lan Khanh

Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 + \frac{1}{11}\right) \left(1 + \frac{1}{19}\right), \end{align*} on which Berndt asked "Is this an isolated result, or are there other identities of this type?". Reb\'ak provided formulas that generate many similar identities and believed that the curious identity is related to the reciprocal of the Landau-Ramanujan constant... (read more)

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