Entry 18

Ramanujan had several expressions for the Riemann zeta function  $$\zeta(s) = \sum_{k=1}^\infty \frac{1}{k^s}$$ especially for \(s=3\), one of which was a rediscovery of Euler's continued fraction (useful for any general series) $$\beta\,\zeta(3) = \cfrac{1}{u_1-\cfrac{1^6}{u_2-\cfrac{2^6}{u_3-\cfrac{3^6}{u_4-\ddots}}}}$$where \(\beta = 1\) and the \( u_n\), starting with \(n = 1\), are, $$u_n = \color{brown}{(n-1)^3+n^3} = (2n-1)(n^2-n+1) = 1, 9, 35, 91, \dots$$Apery found an accelerated version, $$u_n = \color{brown}{n^3 + (n-1)^3 + 4(2n-1)^3} = (2n-1)(17n^2-17n+5) = 5, 117, 535, 1463, \dots$$(notice the sum of cubes) where now \(\beta = \frac{1}{6}\), and established that its rate of convergence was such that \(\zeta(3)\) could not be a ratio of two integers, thus famously proving its irrationality. Ramanujan also gave,$$ \zeta(3) = 1+\cfrac{1}{v_1+\cfrac{1^3}{1+\cfrac{1^3}{v_2+\cfrac{2^3}{1+\cfrac{2^3}{v_3 + \ddots}}}}}$$where the \(v_n\), again starting with \(n = 1\), are given by the linear function,$$ v_n = 4(2n-1) = 4, 12, 20, 28, \dots$$Using an approach similar to Apery's of finding a faster converging version, I found via Mathematica that,$$ \zeta(3) = \cfrac{6}{w_1 + \cfrac{1^3}{1 + \cfrac{1^3}{w_2 + \cfrac{2^3}{1 + \cfrac{2^3}{w_3 +\ddots}}}}}$$where the \(w_n\) are now defined by the cubic function, $$w_n = 4(2n-1)^3 = 4, 108, 500, 1372, \dots$$ Unfortunately, no comparable continued fraction is yet known that proves the irrationality of \(\zeta(5)\).