Loading [MathJax]/jax/output/HTML-CSS/jax.js

Friday, September 23, 2016

Entry 18

Ramanujan had several expressions for the Riemann zeta function  ζ(s)=k=11ks especially for s=3, one of which was a rediscovery of Euler's continued fraction (useful for any general series) βζ(3)=1u116u226u336u4where β=1 and the un, starting with n=1, are, un=(n1)3+n3=(2n1)(n2n+1)=1,9,35,91,Apery found an accelerated version, un=n3+(n1)3+4(2n1)3=(2n1)(17n217n+5)=5,117,535,1463,(notice the sum of cubes) where now β=16, and established that its rate of convergence was such that ζ(3) could not be a ratio of two integers, thus famously proving its irrationality. Ramanujan also gave,ζ(3)=1+1v1+131+13v2+231+23v3+where the vn, again starting with n=1, are given by the linear function,vn=4(2n1)=4,12,20,28,Using an approach similar to Apery's of finding a faster converging version, I found via Mathematica that,ζ(3)=6w1+131+13w2+231+23w3+where the wn are now defined by the cubic function, wn=4(2n1)3=4,108,500,1372, Unfortunately, no comparable continued fraction is yet known that proves the irrationality of ζ(5).

No comments:

Post a Comment