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So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or Michael Somos, but now I am not sure (I don't even know if I trust my memory on the 'S' in the last name) . The page most likely belongs to Jim Cullen and I have been wrongly attributing the open status of these identities to Gourevitch's conjecture (special thanks to Timothy Chow for reverse-engineering my brain and revealing this very unfortunate mental hash collision).

Basically they had experimentally searched the space of identities of the form $\sum_{n=0}^{\infty} \frac{1}{P(n)}$ over a large number of polynomials $P(n)$ and used a tool like Munafo's RIES to see if they matched any known forms. They had struck gold finding some unusual identities involving either $\pi^2$ or $\pi^4$ with quadratic or quartic denominators (I don't remember which). These identities looked like they should be related to the riemann zeta function but the identities had no known proof/explanation at the time.

I want to:

  1. find this website so at least I can edit this question and give the person due credit.

  2. Ask what the state of those identities are. Do they remain open?

I am very happy with the answers and comments this question has generated. I think other people interested in $\pi$ formulas will probably find this as a useful exploration point.

I assume Gourevitch's Conjecture remains open although if anyone had updates on it they could post here or this more trafficked location

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    $\begingroup$ could this be PSLQ? davidhbailey.com/dhbpapers/pslq-comp-alg.pdf , see table 1. $\endgroup$
    – Carlo Beenakker
    Commented May 25 at 20:47
  • $\begingroup$ Thanks for the tip! Let me explore Baileys page. I wonder if that is where I stumbled upon that day. It was an html page not a pdf specifically. $\endgroup$
    – Sidharth Ghoshal
    Commented May 25 at 20:53
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    $\begingroup$ This experimental search is something that I have not attempted. Sorry. $\endgroup$
    – Somos
    Commented May 26 at 0:21
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    $\begingroup$ Could you be thinking of ramanujanmachine.com ? $\endgroup$
    – Robin Houston
    Commented May 26 at 20:54
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    $\begingroup$ @SidharthGhoshal As Jorge Zuniga has commented, Gourevitch's conjecture has now been proved. I have updated my answer to that other MO question accordingly. $\endgroup$
    – Timothy Chow
    Commented May 27 at 21:55

3 Answers 3

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I'm not aware of a perfect match to your description, but here are a few possibilities. Even if they are near-misses, perhaps they will help you with your search.

Jesús Guillera's webpage has a list of various identities involving $\pi$. See also his paper, About a new kind of Ramanujan-type series, Experimental Mathematics 12 (2003), 507–510.

Your mention of $\pi^4$ specifically is reminiscent of Cullen's $\pi^4$ formula. But I see you already know about Cullen's page.

Gert Almkvist is another name that comes to mind. See for example, Some new formulas for $\pi$, by Gert Almkvist, Christian Krattenthaler, and Joakim Petersson, Experimental Mathematics 12 (2003), 441–456.

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  • $\begingroup$ This really might be it. My bad memory may have mixed up Gourevitch's conjecture with Cullen's formula and wrongly attributed it to Cullen's page. I will do a dive on wayback machine and see if theres a version of that page that also says "these identites are open". If not then maybe im fabricating this memory. $\endgroup$
    – Sidharth Ghoshal
    Commented May 26 at 19:29
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4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question, in the EXAMPLES section, I have placed some recently found and proven (Apr. 2024) $\pi^{±4}$ and $\zeta(5)$ new hypergeometric-type identities.

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To follow up on my comment, the specific experimental search for polynomial identities that relate powers of $\pi$ and values of the Riemann zeta function, is in Experimental Evaluation of Euler Sums by Bailey, Borwein, and Girgensohn.
A conjectured identity is $$\zeta(5)=Z_5\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^5{2k\choose k}},\;\;Z_5\in\mathbb{Q},$$ which if correct would nicely complement identities for $\zeta(2)=3\sum_{k=1}^\infty \frac{1}{k^2{2k\choose k}}$, $\zeta(3)=\tfrac{5}{2}\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^3{2k\choose k}}$, and $\zeta(4)=\tfrac{36}{17}\sum_{k=1}^\infty \frac{1}{k^4{2k\choose k}}$.

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