I. Introduction
In Identities Inspired from Ramanujan’s Notebooks, Simon Plouffe recounts how, based on Ramanujan’s,
he found,
and similar ones for other s = 4m+3. On a hunch, and using Mathematica’s LatticeReduce function, I found that,
etc.
II. Functions
If we define,
then Plouffe discovered integer relations between,
for odd s, with s = 3 being,
Eliminating leads to the 3-term equalities in the Introduction. See Chamberland’s and Lopatto’s Formulas for Odd Zeta Values. On the other hand, by defining the function,
I observed integer relations between,
also for odd s, with s = 3 as,
and so on. Eliminating leads to the 4-term equalities in the Introduction.
III. Conjecture
The 4-term equalities have coefficients that are simple except for one term. Recall that,
Conjecture:
“Using the positive case of for s = 4m+3, and the negative for s = 4m+5, then in the equation,
is a rational number.”
The first few for s = {3, 7, 11,…} are while for s = {5, 9, 13,…} are These rationals may have a closed-form expression in terms of Bernoulli numbers, but I do not yet know the exact formulation.