0016: Article 6 (Ramanujan's Pi formulas)

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Ramanujan’s Pi Formulas with a Twist

By Tito Piezas III

Abstract: A certain function related to Ramanujan’s pi formulas is explored at arguments k = {-½, 0, ½} and a conjecture will be given.

I. Introduction

II. Fundamental d with class number h(-d) = 1,2

III. Non-fundamental d with class number h(-d) = 1,2

IV. Higher class numbers h(-d) = 4,6

V. Conclusion: Conjecture

(For an updated summary, also see Ramanujan's Pi Formulas and the Hypergeometric Function.)

I. Introduction

In his 1914 paper, “Modular Equations and Approximations to π”, Ramanujan gave 17 formulas for pi, one of which was,

with the Pochhammer symbol (a)_n = (a)(a+1)(a+2)…(a+n-1). He didn’t provide a proof, just stating that there were “corresponding theories”. It was not until the 1980s work by the Chudnovsky and Borwein brothers that these formulas were placed on a rigorous footing.

In 2003, J. Guillera observed in his paper “Series closely related to Ramanujan formulas for pi” that, for k = ½,

It is easily seen that Ramanujan’s is just the case k = 0. Guillera found that most of Ramanujan’s formulas (as well as those by the Chudnovskys, Borweins, and others) if modified in a similar manner, then evaluated to mln(x), where m and x are rationals. Aptly enough, there is still no proof that the relationships are true, though it holds for arbitrary thousands of decimal digits. What the “corresponding theories” look like remains to be seen.

II. Fundamental d with class number h(-d) = 1,2

In Guillera’s paper, the evaluation mln(x) had a fluctuating m. To have a more well-behaved function, we will use the form,

for some constant k, discriminant d, Pochhammer products h_p, and algebraic numbers {A,B,C}. Define the h_p as,

For k = 0, the formulas below neatly sum to,

However, for k = ½, then,

p = 1:

p = 2:

p = 3:

For p = 1, the denominators of the formulas are the exact values of the negated j-function j(τ), while for p = 2,3, these are for the related modular functions r₂(τ), r₃(τ). (See article, Pi formulas and the Monster group). Note the prime factorizations,

as compared to the function’s respective ln(x), for example,

and one can see an extremely interesting “coincidence”. There is also a similar correspondence between the prime factorization of the denominators of the formulas for p = 2,3, and ln(x).

III. Non-fundamental d with class number h(-d) = 1,2

Let k = ½, then,

p = 1:

p = 2:

p = 3

Evaluated as mln(x), these have a slightly different m from those with fundamental d. If k = 0, then F(0)_d = 1/π.

IV. Higher class numbers h(-d) = 4,6

A. Quadratic irrationals

Guillera pointed out that {A,B,C} may be algebraic and gave a few examples. For fundamental d, we’ll use the smallest with class number h(-d) = 2, and the smallest d = {4u, 3v} with h(-d) = 4, namely d = {15, 68, 39}, respectively. Let {φ, α, β} be the fundamental units,

then,

All evaluate as mln(x) where x is an algebraic number of degree 2 but, as it turns out, the second one does not have the expected m = (1/2)⁴, but has m = (1/2)⁵, which will be relevant for the conjecture in the conclusion.

B. Cubic irrationals

We’ll use the smallest d with class number h(-d) = 3, and the smallest d = {4u, 3v} with h(-d) = 6, namely d = {23, 116, 87}. Let {x,y,z} be the real roots of,

respectively, then,

As expected, the general form is mln(x’) where x’ now is an algebraic number of degree 3. And so on (apparently) for other d with higher class numbers. As usual, all six formulas have F(0)_d = 1/π.

V. Conclusion: Conjecture

Conjecture: “Given the function,

with the Pochhammer products h_p as defined previously, {A,B,C} as algebraic numbers of degree N, and fundamental discriminant d. If,

then,