Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences

Jes\'us Guillera

REVIEW 3 major objections 6 minor 18 references

Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A rational Ramanujan-like series for 1/π^m can be completed to a bilateral sum that is exactly a degree-m trigonometric polynomial, from which the paper derives supercongruences modulo p^{2m+1}.

desk verdict New bilateral completion of Ramanujan-like 1/pi^k series with conjectured supercongruences; the main theorem is plausible but the explicit constants are numerically guessed, and the proved cases rest on cited identities. read the letter →

arxiv 1908.05123 v1 pith:SXO2XMBT submitted 2019-08-14 math.NT

Jesús Guillera This is my paper · ORCID

classification math.NT MSC 11B6533C2011A0711F67

keywords Ramanujan-likeseriesbilateralsemi-terminatingsupercongruenceshypergeometricidentitiesbinomialcongruencesHilbertmodularforms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Ramanujan-like series are hypergeometric sums—infinite sums built from rising factorials—that evaluate to a reciprocal power of π, for example $\sum_{n\ge0}\frac{(\frac12)_n^5}{(1)_n^5}(820n^2+180n+13)(-1/1024)^n=128/\pi^2$. This paper proves that any rational series of this type can be completed to a bilateral sum over all integers, and that the completion is exactly $\frac{v_0\sqrt{(-1)^m\chi_0}}{\pi^m}$ times a trigonometric polynomial of degree $m$ with coefficients $\alpha_k,\beta_k$. Sending the completion parameter to $-p/2$ for odd integers $p$ turns the identity into a finite 'semi-terminating' sum equal to a $\beta$ function times a quadratic-field constant; the paper conjectures that these finite sums obey supercongruences modulo $p^{2m+1}$ for almost all primes, and it verifies the conjecture numerically in all tabulated cases. A few of the supercongruences are proved by reducing them to terminating hypergeometric identities and classical binomial congruences. The paper also records that the quadratic discriminants in its rank-5 table coincide with values of certain L-functions of Hilbert modular forms, and leaves that coincidence unexplained.

What carries the argument

The load-bearing object is the function $$f(x)=$e^{{-i\pi x}}$(\cos\pi x)^{2j+1}\prod_{s_k\ne1/2}\frac{\cos\pi x-\cos\pi s_k}{1-\cos\pi s_k}\sum_{n\in\mathbb Z}A(n,x).$$ It is constructed so that the zeros of $\cos\pi x-\cos\pi s_k$ cancel the poles of the shifted hypergeometric summands, making $f$ holomorphic; it is $1$-periodic because the fractions $s_k$ appear in pairs $s,1-s$; and it grows like $O(e^{(2m+1)\pi|\operatorname{Im}x|})$, which forces the Fourier expansion to have only finitely many terms, up to degree $m$. That finite Fourier identity is the step that converts a Ramanujan-like series into a bilateral identity, then into semi-terminating sums and supercongruences. For the proved cases, the same Fourier reduction is supplemented by terminating hypergeometric identities and classical binomial congruences for primes.

What would settle it

Take Example 4.4, the rank-9 series for $768/\pi^4$. Evaluate both sides of (5) at ten values of $x$ to high precision, solve the resulting $8\times8$ linear system for $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\beta_1,\beta_2,\beta_3,\beta_4$, and check whether the outputs are exactly $-25/6, 8/3, -2/3, 1/6, 0,0,0,0$; any discrepancy beyond rounding error would refute the explicit constants behind the semi-terminating identities and the tabulated $\varepsilon_0$ values. A direct test of the supercongruence conjecture would be to compute the difference in (7) for a tabulated example at a large prime $p$ and verify divisibility by $p^{2m+1}$.

Watch

Extended reading notes

Core claim

The paper's central claim is identity (5): for the shifted summand $A(n,x)$ of a rational Ramanujan-like series for $1/\pi^m$, the normalized bilateral sum $$$e^{{-i\pi x}}$(\cos\pi x)^{2j+1}\prod_{s_k\ne1/2}\frac{\cos\pi x-\cos\pi s_k}{1-\cos\pi s_k}\sum_{n\in\mathbb Z}A(n,x) =\frac{v_0\sqrt{(-1)^m\chi_0}}{\pi^m}\Bigl(1+\sum_{k=1}^m(\alpha_k(\cos2\pi kx-1)+\beta_k\sin2\pi kx)\Bigr),$$ with constants $\alpha_k,\beta_k$ conjectured rational. The proof shows the left-hand side is a holomorphic $1$-periodic function of exponential type $2m+1$, so its Fourier expansion terminates at degree $m$. Letting $x\to-p/2$ with $p$ odd gives the finite semi-terminating identity (6), and the paper conjectures the $p^{2m+1}$ supercongruence (7) for the partial sums $0\le n\le(p-1)/2$; it proves several rank-3 and rank-5 instances by certifiable terminating hypergeometric identities.

Load-bearing premise

The load-bearing assumption is that the constants $\alpha_k$ and $\beta_k$ in identity (5), which the paper determines by solving a finite linear system numerically, have been correctly identified as exact rational values (or exact values times $i$); the paper states this identification without giving an exact proof for the tabulated examples.

Editorial extensions

If this is right

Every rational Ramanujan-like series for $1/\pi^m$ has a bilateral completion whose Fourier polynomial has degree exactly $m$, producing new finite identities at half-integer shifts.
The conjectured supercongruences (7) would hold modulo $p^{2m+1}$ for all odd primes outside a finite exceptional set, with sign and power determined by the series' quadratic discriminant and rank.
For integer shifts the same mechanism yields integer-shift supercongruences modulo $p^{2m+1}$, and adding one further term gives strengthened versions modulo $p^{2m+2}$.
The proved rank-3 and rank-5 cases show the route is effective: terminating hypergeometric identities plus classical binomial congruences are enough to certify the pattern in concrete instances.
If the rationality conjecture for the Fourier coefficients $\alpha_k,\beta_k$ is correct, the semi-terminating constants in (6) are rational, so the supercongruence conjecture is a statement about genuinely arithmetic numbers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If identity (5) is as general as the proof suggests, the same Fourier-termination argument should extend by analytic continuation to the divergent ($|z_0|>1$) cases in the tables, yielding supercongruence predictions there.
The recorded match between the rank-5 discriminants and Hilbert modular L-function values suggests the quadratic fields in the supercongruences are controlled by the same hypergeometric motives; proving this would require linking the finite Fourier completion to modular-form arithmetic.
The finite Fourier structure of (5) gives an algorithmic rationality test for candidate series: compute the first $2m$ Fourier coefficients numerically and check whether they are rational (or rational times $i$); a failure would rule out a Ramanujan-like series for $1/\pi^m$.
Computing the coefficients $\alpha_k,\beta_k$ exactly for a rank-7 or rank-9 example beyond the tables would test the rationality conjecture and, if it failed, would force a revision of the explicit semi-terminating constants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is identity (5): a bilateral completion of rational Ramanujan-like series that, at x = -p/2, produces semi-terminating identities and a whole family of conjectured p^{2m+1} supercongruences (7). That is a real idea, and the connection to Asai L-functions via the epsilon_0 table is worth taking seriously even though it is only an observation. The paper is also honest about what is proved versus conjectured: Section 4 is explicitly flagged as conjectural, and Section 5 proves several rank-3 and rank-5 B-supercongruences using terminating hypergeometric identities plus Morley and Wolstenholme. Those proofs look correct to me on inspection, and Remark 5.2 gives a credible Zeilberger route for certifying the identities, so the proved examples are not floating on trust.

The soft spots are real but localized. The proof of (5) is sketchy: the growth bound O(e^{(2m+1)pi|Im x|}) is asserted without derivation, and the termination at k=m is the load-bearing step. More importantly, the explicit alpha_k and beta_k are obtained by numerically solving a linear system and then an unverified "easy" rational identification. The reader's stress-test note is right: if those numbers are merely good rational approximations, the semi-terminating formulas in Examples 4.1-4.4 and the tabulated epsilon_0 are wrong, and the numerical support for (7) collapses. That is a genuine gap, not a manufactured one. It is also fixable: one could verify the coefficients exactly by checking the Fourier expansion at enough rational points with rigorous interval arithmetic, or by proving the finite Fourier structure directly. The A-side congruence (12) is not new, being equivalent to Zudilin-type supercongruences, so the real contribution is the B-side framework.

I disagree with any reading that calls this circular. The conjectures are not fitted to data; the proved examples are derived from independently known identities, and the numerical checks are presented as checks. The self-citation is fine.

Who is this for? People working on Ramanujan-type series, supercongruences, and hypergeometric motives. A serious referee should engage with it, mainly because the bilateral identity is novel and likely to be useful even if some of the conjectured constants need adjustment. I would not desk-reject. My recommendation: send to a specialist in hypergeometric congruences, and push the author to supply exact verification of the alpha_k, beta_k values or to state clearly that they are conjectural throughout Section 4.

Referee Report

3 major / 6 minor

Summary. The paper constructs bilateral series associated with rational Ramanujan-like series for negative powers of π. It introduces functions A(n,x) and B(n,x), forms a periodic holomorphic function f(x) with a cosine prefactor, and states identity (5) asserting that f(x) equals v0√((−1)^m χ0)/π^m times a finite trigonometric polynomial of degree m with constants α_k, β_k (conjectured rational). From the limit x → −p/2 the author derives a semi-terminating identity (6) and conjectures supercongruences (7) modulo p^{2m+1}. Section 4 gives unproved examples with explicit α_k, β_k, r0, and ε0. Section 5 proves several B-supercongruences for rank-3 and rank-5 series using terminating hypergeometric identities quoted from [6] together with Morley's and Wolstenholme's congruences. The paper ends with tables of known rational series and a speculative observation linking the ε0 values to Asai L-functions in [4].

Significance. If identity (5) is rigorously established, the paper would provide a uniform bilateral completion for rational Ramanujan-like series and a new mechanism for producing B-supercongruences with modulus p^{2m+1}. The proved supercongruences in Section 5 are concrete and appear correct; reducing them to terminating hypergeometric identities and classical congruences is a clean and effective strategy. The conjecture (7), with explicit ε0 values, is falsifiable and the tabulated data constitute a useful experimental resource. The link to Asai L-functions, though speculative, is intriguing. However, the proof of the central identity is sketchy, and the exact values of α_k and β_k used in Section 4 are obtained by numerical identification rather than proof, so the rigorous core of the paper is currently limited to Section 5.

major comments (3)

[Section 2, proof of identity (5)] The proof of (5) hinges on the unproved estimate f(x)=O(e^{(2m+1)π|Im(x)|}). This bound is stated without derivation, and a reader cannot check how the infinite sum A(n,x) and the prefactors behave near infinity or how the pole cancellations contribute to the growth. Since this estimate is what forces the Fourier expansion to contain only the modes up to k=m, identity (5) is not fully proved as written. Please supply the necessary estimates (or a precise reference) and spell out how the bound yields the exact trigonometric form in (5).
[Section 2, paragraph after (5); Section 4] The constants α_k and β_k are determined by 'solving numerically the linear system' and then 'is easy to identify the exact rational values they are.' This is not a proof of exactness. These constants enter directly into the semi-terminating constant r0 in (6) and into the tabulated ε0 and the numerical checks of conjecture (7) in Examples 4.1–4.4 and Tables 1–5. The author should either prove the exact values by an exact method (for example, by differentiating (5) at x=0 and using exact arithmetic, or by evaluating at enough special points and solving the linear system exactly) or explicitly state that the listed α_k, β_k, r0, and ε0 are conjectural. As written, the sentence invites the reader to treat numerical approximations as exact identities.
[Section 5, use of terminating identities from [6]] The proofs of Examples 5.1–5.7 start from terminating hypergeometric identities quoted from [6] (for instance, (24), (30), (35), (40), (45), (50)). Although Remark 5.2 states that Zeilberger's algorithm can verify such identities, no recurrences, initial values, or certificates are shown, so the displayed identities cannot be checked by the reader. Since these identities are the foundation of the proofs in this section, please either reproduce the relevant derivation from [6] with the exact substitutions and limits, or include an appendix with the Zeilberger recurrences and initial values. Otherwise the 'proved' examples rest on unverifiable input.

minor comments (6)

[Section 3 and Introduction] The word 'semicongruences' appears where 'supercongruences' is clearly intended; please correct the terminology consistently.
[Equation (17)] The right-hand side of (17) is typographically ambiguous: '= 8 8p (2p − 1)!/(p − 1)!' should probably read '= 8/8^p (2p − 1)!/(p − 1)!' or similar. Please clarify the intended denominator.
[Example 5.6, congruence (44)] The congruence (44) is missing the modulus at the end; it should read '(mod p^5)'.
[Example 5.7] The word 'identy' should be 'identity' in the proof of Example 5.7.
[Section 6, Tables] The tables do not state how the ε0 values were obtained. A short note on the provenance (numerical evaluation of (6), conjectural pattern, or exact derivation) would improve reproducibility and clarify the status of each entry.
[Section 6, Intriguing observation] The final observation would be more useful if the author explicitly identified which entries in [4, Table 5.2] match the ε0 values in Table 3; as written, the correspondence is not checkable from the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: identity (5) is proven from (2) via a terminating Fourier expansion; the numerical identification of the Fourier constants is a correctness risk, not a circular step.

full rationale

The derivation chain is not circular. The central identity (5) is proved from the original Ramanujan-like identity (2): the displayed function f is holomorphic, 1-periodic, and has exponential growth O(e^{(2m+1)pi|Im x|}), so its Fourier expansion terminates at degree m; evaluating at x=0 (where A(n,0)=0 for n<0 because of the (1)_n denominator) identifies the constant term as v0 sqrt((-1)^m chi0)/pi^m, and the alpha_k, beta_k are the normalized Fourier coefficients. The subsequent semi-terminating formula (6) is obtained by taking a limit in (5), not by re-fitting the data asserted in (7). The alpha_k, beta_k in Section 2 are found by solving the linear system for the provably existing Fourier coefficients; this is a numerical-identification step, and the paper explicitly labels Section 4 'All the content of this section is conjectured.' The supercongruence conjecture (7) is a genuinely new p-adic statement checked for many primes; its modulus and RHS are fixed by (6) and by the exact constants used, not chosen to match the checked sums. Section 5's proved examples use terminating hypergeometric identities from [6] and classical congruences of Morley, Wolstenholme, and Zudilin; the [6] identities are WZ/Zeilberger-certifiable (Remark 5.2) and are not equivalent to the congruences being proved. Thus there is no step in which an output is built into an input by definition, by fitting, or by an unverified self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the exact values of the Fourier coefficients and the discriminant constants; these are not pulled from thin air but are numerically fitted and then conjectured to be exact. No new physical or mathematical entities are introduced. The proofs of the Section 5 cases avoid these fitting parameters by using exact terminating identities, so the fitted values affect the conjecture, not the proved results.

free parameters (2)

alpha_k, beta_k (Fourier coefficients in (5)) = alpha1=-14/3, alpha2=2, beta1=0, beta2=0 (Example 4.1); alpha1=-79/2, alpha2=23/2, beta1=5i/2, beta2=-3i/2 (Example 4.2)
Determined by numerically solving the linear system mentioned after (5); the paper identifies them as exact rational (or iQ) values without proof, and their exactness is conjectural.
epsilon_0 (discriminant in conjecture (7)) = e.g., 1 (Example 4.1), -7 (Example 4.2); tabulated in Tables 1-5
Conjectured constants in the supercongruence (7), numerically verified for many primes according to the abstract; not derived except in a few proved cases in Section 5.

assumptions (4)

domain assumption The rational Ramanujan-like series identities (2) hold for the series used in Sections 4 and the tables; the paper notes that most of these series remain unproved.
The bilateral identity (5) is derived from (2), so the starting point is conjectural for the unproved series; the final supercongruences are finite statements that could in principle be checked independently.
domain assumption The periodic holomorphic function f(x) defined in the proof of (5) satisfies the growth bound O(e^{(2m+1)π|Im x|}), which forces its Fourier expansion to terminate at degree m.
This bound is asserted, not derived; it is structurally necessary for the trigonometric-polynomial form of (5).
standard math The terminating hypergeometric identities from [6] (eqs. 21, 24, 27, 35, 36) used in Section 5 are correct as proved by the WZ method in that paper.
They underpin the proofs of Examples 5.1, 5.3, 5.4, 5.5, and 5.6; the paper cites [6] but does not reproduce the certificates.
standard math Morley's congruence (28), Wolstenholme's congruence (43), and congruence (48) from [17] are valid.
They are the final reduction steps in the Section 5 proofs.

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Cite this review

Pith. "Pith review of Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences." pith.science (2026). https://pith.science/paper/SXO2XMBT

@misc{pith2026190805123,
  author       = {Pith},
  title        = {Pith review of: Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXO2XMBT}},
  note         = {Machine review of arXiv:1908.05123}
}

abstract

We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In addition we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Guillera , More hypergeometric identities related to Ramanujan-type serie s, The Ramanujan J

J. Guillera , More hypergeometric identities related to Ramanujan-type serie s, The Ramanujan J. 32, (2013), 5–22

work page 2013

[2] [4]

Special hypergeometric motives and their $L$-functions: Asai recognition

L. Dembel ´ e, A. Panchishkin, J. Voight and W. Zudilin , Special Hypergeometric Motives and their L-Functions: Asai recognition, Preprint (2019): https://arxiv.or g/abs/1906.07384

work page Pith review arXiv 2019

[3] [1]

Almkvist and J

G. Almkvist and J. Guillera , Ramanujan-like series for 1 /π 2 and String Theory, Exp. Math. 21, (2012), 223–234

work page 2012

[4] [2]

Baruah, B.C

N.D. Baruah, B.C. Berndt and H.H. Chan , Ramanujan’s series for 1 /π : A survey, The Amer. Math. Monthly 116 (2009), 567–587

work page 2009

[5] [3]

Chisholm, A

S. Chisholm, A. Deines, and H. Swisher , Recent advances for Ramanujan type supercongru- ences, Contemporary Mathematics (2013)

work page 2013

[6] [5]

Guillera , A matrix form of Ramanujan-type series for 1 /π , Gems in Experimental Mathematics: Contemp

J. Guillera , A matrix form of Ramanujan-type series for 1 /π , Gems in Experimental Mathematics: Contemp. Math. 517, (2010), 189–206

work page 2010

[7] [7]

Guillera , WZ pairs and q-analogues of Ramanujan series for 1 /π (with an appendix by Wadim Zudilin), Journal of Diﬀerence Equations and Applications 24, 1871– 1879

J. Guillera , WZ pairs and q-analogues of Ramanujan series for 1 /π (with an appendix by Wadim Zudilin), Journal of Diﬀerence Equations and Applications 24, 1871– 1879

work page

[8] [8]

Divergent

J. Guillera & W. Zudilin , “Divergent” Ramanujan-type supercongruences, Proc. Amer . Math. Soc. 140 (2012), 765–777

work page 2012