Identities inspired by Ramanujan Notebooks (part 2)
by Simon Plouffe

April 2006

These are new findings related to the ones found in 1998. I had the idea that maybe the sums in the first note I wrote were looking in the bad direction. They are good yes but maybe there is better.  Actually there is, instead of considering powers of exp(2π) I should consider powers of exp(kπ).

After a few experiments with Maple (computer algebra system) and PSLQ (integer relations algorithm) here are the findings and remarks.

- I could go up to Zeta(39) and powers of exp(30π) after that the numbers are too close together for any computation,  1/exp(30π) is of the order of 10^-40 and even with 2000 decimal digits I get an error.

- The powers of exp(kπ)±1 are related to the divisors of a number.

- The numbers that appear to be expressible are so far : log(A) where A is an integer or simple alebraic like arccosh(1/2) or arctanh(2/3), ζ(2n+1), π²ⁿ⁺¹, ζ(2n+1) and some formulas involving lnΓ(1/4) and ln(π) but not separately.

Formulas for odd powers of π and ζ(2n+1)

That one is the fastest converging series so far with 5.45 digits for each n.

Other formulas

The last formula for Zeta(5)  is interesting since the lowest term gives a rate of convergence of the order of 1/exp(3π) which is 4.09 decimal digits.

References
[1] Berndt, Bruce,  Ramanujan Notebooks (volumes I to V), Springer Verlag.

[2] Plouffe, Simon, Identities inspired from Ramanujan Notebooks II, 1998, unpublished notes.

[3] Finch, Steven R., Mathematical Constants, Encyclopedia of Mathematics and its Applications, 94 (2003).

[4] Flajolet, Philippe. Recent articles from INRIA.

I take the occasion to thank Bruce Berndt for the remarkable piece of work he did with these collected works of S. Ramanujan.