0027b: Part 6b, Complex series for pi

Part 6b: Complex Series for 1/Pi using all Roots of the Hilbert Class Polynomial

by Tito Piezas III

I. Hilbert Class Polynomial

II. Conjecture

III. Other Series

I. Hilbert Class Polynomial

It is well-known that the j-function j(τ), for the form τ = (1+√-d)/2 or τ = √-d, is an algebraic number with degree k equal to the class number h(-d). Its minimal polynomial is called the Hilbert class polynomial, H_d.

Of course, the smallest degree such that H_d has complex roots is class number h(-d) = 3 with the smallest d as d = 23. Let,

A_u be any root of x³- 1334x²+ 69069x- 1216171 = 0 (eq.1)

B_v be any root of y³- 27y²+ 56y - 37 = 0 (eq.2)

C_w be any root of z³+ 155z²+ 650z + 23375 = 0 (eq.3)

All three equations have only one real root and, following the root numbering system of Mathematica, we have C₁ as the real root, and C₂ and C₃ as complex conjugates. (Likewise for A_u and B_v.) Explicitly,

C₁³ = j((1+√-23)/2) ≈ -3.4932 x 10⁶

Then, we have,

where,

The first series uses all real {A, B, C}, but the last two use the complex roots of (eqns.1, 2, 3), where their subscripts follow the root numbering system of Mathematica. For d = 47 which has class number h(-d) = 5, using all five roots of the Hilbert class polynomial H₄₇, it can be shown there are five similar equalities. And so on, apparently, for other d.

II. Conjecture

“Given {A, B, C} such that,

where {A, B, C} are algebraic numbers and P(x), P(y), P(z) their minimal polynomials. Let C_w be any complex root of P(z) = 0. Using the appropriate complex roots of the same minimal polynomials P(x), P(y) equated to zero, then,

where m is a rational number.”

As usual, the h_p are the factorial ratios, or Pochhammer symbol products (a)_n,

Note: The series in the section below no longer use the Hilbert class polynomial.

III. Other Series

In “Divergent” Ramanujan-Type Supercongruences, J. Guillera and W. Zudilin discovered the first series for 1/π with complex coefficients,

which involves Pochhammer symbols for p = 4. Since then, cases p = 2, 3 have been found by H.H. Chan, J. Wan, and W. Zudilin, in “Complex Series For 1/π”. However, there is a case p = 3 found by this author,

where i is the imaginary unit. Equivalently, if we wish to simplify the terms as well as express the complex denominator on the RHS as a cube,

where the conjugate -i can also be used. It is the only known convergent series using only rational complex numbers discovered so far, and was found using the formulas for p = 3 in the previous article, Part 6: General forms for Ramanujan's Pi Formulas. The denominator on the RHS can also be used in a complex series for the complete elliptic integral K(k₂) as,

See next article for more details.

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You can email author at tpiezas@gmail.com.

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