Friday, March 01, 2013

[eefzdetg] Underlying approximation to Ramanujan pi formula

The Chudnovsky formula for pi, (12 * sum(n=0, Infinity, (-1)^n * (6 * n)! * (13591409 + 163 * 3344418 * n) / ((n!)^3 * (3 * n)! * (640320^3)^(n + 1 / 2))))^-1 , when truncated to one term, 53360 * sqrt(640320) / 13591409, it has an error of -(10^-13.2). However, underlying the formula is the Heegner number -163, from which we can get a more accurate approximation ln(640320^3+744)/sqrt(163), which has an error of 10^-30.7. For completeness, 640320 = 2*2*3*((3*7*11)^2-1) and 3344418 = 3*7*11*14478, both containing 231.

Ramanujan's formula (sqrt(8) / 99^2 * sum(n=0, Infinity, (4 * n)! * (1103 + 26390 * n) / ((n!)^4 * 396^(4 * n))))^-1 , is structurally similar to Chudnovskys'.  When truncated to one term, 99^2/(2*1103*sqrt(2)), it has an error is 10^-7.1 . However, does it have a more accurate underlying approximation analogous to the Heegner number?

Update: answer: ln(396^4-104)/sqrt(58) has error 10^-18.0, and the coefficient 26390 = 58 * 455. Fundamental discriminant -4*58= -232 and class number 2. Is the relevant analogous factorization 396 = 2*2*((2*5)^2-1) ?