# [Calculation 19] Even values of the Zeta Function

(revision of 2012)  In this post, we evaluate even values of the Riemann zeta function. Calculation 19. For $n\in \N$, we have $$\zeta(2n) = \frac{|B_{2n}|2^{2n-1} \pi^{2n}}{(2n)!},$$ where $B_n$’s are Bernoulli numbers.   Proof. We have the general formula for … Continue reading

# [Calculation 18] Some Useful Formulas from the Jacobi Triple Product

In this post, we introduce some useful formulas, which can be proved by the Jacobi Triple Product [2]. As before, we always denote $f$ as the Ramanujan Theta Function, which is defined in [3].   Corollary. If $|q|<1$ then, \begin{align*} … Continue reading

# [Calculation 16] Ramanujan ‘s 1ψ1 (1-psi-1) Summation Formula

In this post, we will introduce one of the famous formulas discovered by Ramanujan, which is called Ramanujan’s ${}_1\psi_1$ Summation Formula. It was first introduced by Hardy, and he called it as “a remarkable formula with many parameters”. The first … Continue reading