[Paper Review] Kind of proofs of Ramanujan-like series

Jesús Guillera|arXiv (Cornell University)|Mar 6, 2012

Advanced Mathematical Identities22 references3 citations

TL;DR

This paper classifies and analyzes different proof types for Ramanujan-like series for $1//pi$ and $1//pi^2$, introducing new hypergeometric transformations and conjecturing modular-type relations essential for proving algebraic values in $1/\/pi^2$ series. It presents a novel non-rational $z$-value ($z = -27/\phi^3$) for a conjectured $1/\pi^2$ series with fast convergence and links it to special functions and L-functions via PSLQ-based integer relations.

ABSTRACT

We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/π^2.

Motivation & Objective

• To systematize and extend known proof techniques for Ramanujan-type series for $1/\pi$ and $1/\pi^2$.
• To identify missing modular-type structures necessary for proving $1/\pi^2$ series with non-rational $z$-values.
• To explore the role of hypergeometric transformations and analytic continuation in deriving new series for $1/\pi^2$.
• To investigate the connection between conjectured series and special values of $L$-functions and zeta functions via integer relation algorithms.
• To provide a framework for understanding divergent series as convergent sums of residues, inspired by supercongruence patterns.

Proposed method

• Utilizes $q$-parametrization of hypergeometric series and modular functions, expressing $z$, $a(z)$, and $b(z)$ as functions of $\tau$ via $q = e^{-\pi\tau}$.
• Applies modular equations of various orders (e.g., septic, duality, third-order) to derive algebraic relations for $z$-values in Ramanujan-type series.
• Employs hypergeometric transformations, such as those involving $\,_{2}F_{1}$ and $\,_{5}F_{4}$, to relate different series forms and enable translation between series types.
• Uses Zudilin’s translation method to transform known series into new ones with rational $\tau_2/\tau_1$ ratios, preserving convergence and structure.
• Applies the PSLQ algorithm to detect integer relations among special values of hypergeometric functions and $1/\pi^2$, leading to conjectured identities.
• Interprets divergent series as sums of residues via analytic continuation and the reflection formula $\cos(\pi s)$, linking them to zeta and $L$-values.

Experimental results

Research questions

• RQ1 What modular-type structures are missing that would allow a rigorous proof of Ramanujan-like $1/\pi^2$ series with non-rational $z$-values?
• RQ2 How can hypergeometric transformations be systematically used to generate new Ramanujan-type series for $1/\pi^2$?
• RQ3 What is the role of functional relations and conjectured identities (e.g., Conj. 3.2, Conj. 3.3) in explaining algebraic values in $1/\pi^2$ series?
• RQ4 Can divergent series with non-rational $z$ be interpreted as convergent sums of residues, and do they yield known special values like $\zeta(3)$ or $L_5(3)$?
• RQ5 What is the significance of the non-rational $z = -27/\phi^3$ in the context of $1/\pi^2$ series, and how does it relate to modular forms and $L$-functions?

Key findings

• A new conjectured $1/\pi^2$ series is proposed with $z = -27/\phi^3$, where $\phi = \left(\frac{\sqrt{5}-1}{2}\right)^5$, and it converges rapidly to $3/\pi^2$.
• The minimal polynomial of the $z$-value is $P(z) = z^2 + 36828z - 729$, and the other root $z = (-3/\phi)^3$ is conjectured to correspond to a second series.
• Using PSLQ, a new conjectured series for $36/\pi^2$ is derived with coefficients involving $\phi$, and it satisfies mosaic supercongruence patterns.
• The divergent series $\sum_{n=1}^{\infty} \cdots$ is interpreted as a convergent sum of residues, yielding $95232\zeta(5)$, with the imaginary part $-160\pi^5 i$ vanishing upon inclusion of $\cos(\pi s)$.
• The sum of the inverse series is conjectured to equal $\frac{1125}{4}\sqrt{5}L_5(3) - 448\zeta(3)$, enabling high-precision computation of $L_5(3)$ via fast convergence.
• The paper conjectures that undiscovered modular equations and functional relations (e.g., Conj. 3.2, Conj. 3.3) could unify and explain the algebraic values in $1/\pi^2$ series.