arxiv: 2604.03327 · v2 · submitted 2026-04-02 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Series for 1/π arising from Cauchy product

Roman Le Lan

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:04 UTC · model grok-4.3

classification 🧮 math.NT

keywords 1/π seriesCauchy producthypergeometric transformationsSun conjecturepi identitieshypergeometric seriesbinomial series

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The pith

A series for 1/π conjectured by Sun is evaluated exactly using the Cauchy product of hypergeometric series.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the exact value of an infinite series for 1 over pi that was previously only conjectured. The evaluation proceeds by forming the Cauchy product of two hypergeometric functions and applying transformations to simplify the result to a known multiple of 1/π. Two further series of similar form but with cubic polynomials in the general term are also derived by the same steps. The method appears reusable for additional identities of this type, as indicated by a table of further cases.

Core claim

The paper proves that a particular series conjectured by Sun sums to 1/π. The proof combines the Cauchy product with hypergeometric series transformations. From this, two analogous series involving polynomials of degree three are obtained, and further identities are listed in a table.

What carries the argument

Cauchy product applied to hypergeometric series, followed by transformations that identify the sum with a multiple of 1/π.

If this is right

The conjectured series equals 1/π.
Two new series with degree-3 polynomials also equal 1/π.
Additional similar series can be proved by the same technique.
These identities provide new representations for 1/π.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may generalize to other hypergeometric series conjectures for constants like pi.
Such proofs could help classify or generate families of pi-series without relying on numerical guessing.
It connects to broader work on hypergeometric identities in number theory.

Load-bearing premise

The hypergeometric transformations remain valid for the parameters in Sun's series without any singularities or convergence issues inside the radius.

What would settle it

Direct numerical summation of the series terms to high precision, checking whether partial sums converge to 1/π within the expected truncation error.

read the original abstract

In this note, we evaluate a series for $1/\pi$ conjectured by Sun. Our proof uses the Cauchy product and hypergeometric transformations. From this result, we derive two additional analogous series for $1/\pi$ involving polynomials of degree $3$. Further identities can be proved using our method; these are presented in a table at the end of the note.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Proves Sun's conjectured 1/π series via Cauchy product of hypergeometrics but needs explicit check on radius and singularities at the evaluation point.

read the letter

This note proves one of Sun's conjectured series for 1/π by taking the Cauchy product of two hypergeometric series and then applying standard transformations to get the closed form. It also derives two new series with cubic polynomial coefficients from the main result. The work does a solid job of turning the conjecture into a theorem with elementary steps that rely on known properties of hypergeometric functions. No parameters are fitted, and the argument builds directly from published identities. Adding the table of further identities is helpful for anyone collecting these kinds of formulas. The main soft spot is the application of the transformations at the specific point in the complex plane. The stress test points out that the combined series might hit the boundary of the convergence disk or encounter singularities for the parameters in Sun's series. The paper should include a brief check or justification for why the equality holds there, perhaps via analytic continuation. If that's missing, it's a minor but real gap that a referee could flag. This is for people who follow work on Apéry-like series and explicit representations of 1/π. A reader who wants new identities or a template for proving similar conjectures will find it useful. The thinking is clear and the method is reproducible from the description. I recommend sending it to peer review. The result is incremental, but the proof is direct enough that checking the details is worthwhile and could be done without much effort.

Referee Report

1 major / 2 minor

Summary. The manuscript evaluates a series for 1/π conjectured by Sun by forming the Cauchy product of two hypergeometric series and applying standard transformations to obtain a closed form. From the resulting identity, two further series for 1/π are derived, each involving a cubic polynomial in the numerator, and a table of additional identities obtainable by the same method is presented.

Significance. If the central derivation is valid, the work supplies a direct, non-numerical confirmation of Sun’s conjecture together with a reusable technique based on the Cauchy product and hypergeometric identities. The absence of fitted parameters and the explicit derivation of two new cubic-polynomial series constitute genuine additions to the literature on π-series.

major comments (1)

[Proof of the main identity] The main evaluation step (immediately after the Cauchy product is formed) invokes hypergeometric transformations without an explicit verification that the target argument lies inside the disk of absolute convergence for the specific rational parameters appearing in Sun’s series. Because the combined radius of the product series places the evaluation point at or on the boundary, the substitution requires either a direct radius calculation or a justification via Abel summation or analytic continuation; this step is load-bearing for the claimed equality.

minor comments (2)

[Final table] The table of further identities would benefit from a brief indication, for each entry, of which transformation (Pfaff, Euler, or contiguous) is applied.
[Section 2] Notation for the hypergeometric parameters should be introduced once at the beginning of the proof rather than re-defined inline.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit convergence justification in the central step of the proof. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The main evaluation step (immediately after the Cauchy product is formed) invokes hypergeometric transformations without an explicit verification that the target argument lies inside the disk of absolute convergence for the specific rational parameters appearing in Sun’s series. Because the combined radius of the product series places the evaluation point at or on the boundary, the substitution requires either a direct radius calculation or a justification via Abel summation or analytic continuation; this step is load-bearing for the claimed equality.

Authors: We agree that an explicit verification is required for rigor. The transformations applied are standard hypergeometric identities that hold by analytic continuation at the boundary point for the specific rational parameters in Sun’s series (as the resulting closed form matches the known value of the series). In the revised manuscript we will insert a direct radius-of-convergence calculation for the transformed series, confirming that the evaluation point lies inside the disk of absolute convergence, together with a brief appeal to Abel summation to justify the boundary case if needed. This addition will be placed immediately after the Cauchy-product step. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard external identities

full rationale

The paper evaluates Sun's conjectured series for 1/π by forming the Cauchy product of two hypergeometric series and applying known transformation identities (Pfaff, Euler, or contiguous relations). These operations are independent of the target result; the transformations are invoked as previously established properties of hypergeometric functions, not derived within the paper or fitted to the specific series. No parameters are adjusted to the output, no self-citations bear the central claim, and the argument remains self-contained against external benchmarks. The derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies only on standard, previously established facts about hypergeometric series and their transformations; no new constants, entities, or fitted parameters are introduced.

axioms (2)

standard math Standard transformation identities for hypergeometric functions hold for the parameters in Sun's series
Invoked to collapse the Cauchy product into a closed form.
domain assumption The series converge absolutely in the region where the transformations are applied
Required for the Cauchy product to equal the product of the sums.

pith-pipeline@v0.9.0 · 5338 in / 1203 out tokens · 52342 ms · 2026-05-13T20:04:49.172491+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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P(z) expressed as product of two 2F1 functions, then Euler transformation and second Cauchy product to obtain central binomial series
IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaInverseRS unclear

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Series equals 3√6/(2π) via Guillera Ramanujan-type formula

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Almkvist and A

H. Almkvist and A. Aycock,Proof of some conjectured formulas for1/πby Z.-W. Sun, 2011. Preprint athttps://arxiv.org/abs/1112.3259

work page arXiv 2011
[2]

Guillera,A method for proving Ramanujan series for1/π, Ramanujan J.52(2018), no

J. Guillera,A method for proving Ramanujan series for1/π, Ramanujan J.52(2018), no. 2, 421–431

work page 2018
[3]

Sun,List of conjectural series for powers ofπand other constants, 2011

Z.-W. Sun,List of conjectural series for powers ofπand other constants, 2011. Preprint at https://arxiv.org/abs/1102.5649

work page arXiv 2011
[4]

Sun,Conjectures and results onx 2 modp 2 with4p=x 2 +dy 2, Number theory and related areas, 2013, pp

Z.-W. Sun,Conjectures and results onx 2 modp 2 with4p=x 2 +dy 2, Number theory and related areas, 2013, pp. 149–197

work page 2013
[5]

Sun,New type series for powers ofπ, J

Z.-W. Sun,New type series for powers ofπ, J. Comb. Number Theory12(2020), no. 3, 157– 208. a b q α a b q α 3−1 1 2 3 √ 6 2 18150 1433− 1 3024 2592 √ 3 50 19− 9 16 32 √ 3 3 16854 985− 1 5002 750000 √ 267 7921 17 1− 1 16 16 √ 51 17 75 7 2 27 81 √ 3 4 162 19− 1 80 32 √ 3 363 38 4 125 17 √ 15 2 150 13− 1 1024 6144 √ 123 1681 Table 1.Other series of the form (...

work page 2020