**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.

Posted by gerrymrt on June 28, 2012 at 8:43 am

Combining both cases s=4m+3 and s=4m+5 using (-1)^s confirms the connection between the Bernoulli numbers and your V1(s),V2(s) as follows:

Bernoulli[n_] := 4n ((-1)^n V1[1-2n]- 2^n V2[1-2n])/((-1)^n-2^n)

BernoulliB[2n] == Bernoulli[n] for n>1

Posted by gerrymrt on June 28, 2012 at 9:08 am

The rational F(s) relationship between V1,V2 and the Odd Zeta values can also be combined as follows

F[s_] := (2^s V1[2 s + 1] – (-1)^s V2[2 s + 1] +

1/2 (2^s – (-1)^s) Zeta[1 + 2 s])/(Sqrt[2] Pi^(2 s + 1))

giving the sequence (first 10 values):

{1/24, 1/270, 41/37800, 19/103950, 29/714420, 5017/638512875, 707339/434188755000, 1069039/3275571048750,

30021059/451339210822500, 13097369/974482387003125}

Posted by tpiezas on June 28, 2012 at 4:32 pm

Yes, I inserted the even zeta values to reduce the size of the denominators.