Basel problem: a physicist's solution

Z.K. Silagadze

Basel problem: a physicist's solution

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that the Basel sum equals $\pi^2/6$: it derives the needed trigamma reflection formula from a differential equation, then extends the same route to all even zeta values.

desk verdict A fresh pedagogical proof of ζ(2)=π²/6 via a Coulomb-force analogy and an ODE proof of the digamma reflection formula; the delicate ε-limit step is informal but fixable, and the paper earns its place as an expository note. read the letter →

arxiv 1908.07518 v1 pith:NOIGNY4H submitted 2019-08-20 math.HO math-phmath.MP

Z.K. Silagadze This is my paper · ORCID

classification math.HOmath-phmath.MP MSC 11M0633B1511B68

keywords BaselproblemRiemannzetafunctiontrigammadigammareflectionformulaprincipal-valueidentitytangentnumbersevenvalues

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

This paper offers a short proof that the Basel sum $\sum_{n=1}^{\infty}1/n^2$ equals $\pi^2/6$, motivated by a physical picture: a regularized potential whose gradient is an inverse-square force acting on a unit charge at $x=1/2$ from charges at the positive integers. The force calculation turns the sum into one third of the trigamma function at $1/2$, so the missing ingredient is the reflection formula for the trigamma function. The paper proves that reflection formula by showing that $\varphi(x)=\psi(x)-\psi(1-x)$ obeys the differential equation $\varphi'=\pi^2+\varphi^2$, which with $\varphi(1/2)=0$ gives $\varphi(x)=-\pi\cot\pi x$. The same scheme yields all even zeta values through tangent numbers and gives a recurrence for them.

What carries the argument

The load-bearing object is the function $\varphi(x)=\psi(x)-\psi(1-x)$ and the ordinary differential equation $\varphi'(x)=\pi^2+\varphi(x)^2$ that it is shown to satisfy. The derivation uses the principal-value identity $\frac{1}{z\pm i\epsilon}=\mathcal P\frac{1}{z}\mp i\pi\delta(z)$ to multiply two representations of $\varphi(x)$, then changes variables and interchanges integrations to obtain $\varphi^2+\pi^2=-\int_0^\infty y^{-x}\ln y/(1-y)\,dy$; differentiating the principal-value representation gives $\varphi'$ as the same integral, so the ODE follows. Its solution under $\varphi(1/2)=0$ is $-\pi\cot\pi x$, exactly the digamma reflection formula, and one more derivative gives the trigamma reflection formula that closes the Basel computation.

What would settle it

Take a non-special point such as $x=0.37$, evaluate $\varphi(x)=\psi(x)-\psi(1-x)$ to high precision, and check numerically whether $\varphi'(x)$ equals $\pi^2+\varphi(x)^2$; any deviation beyond roundoff would refute the central differential equation. Alternatively, evaluate the double integral in equation (22) with a small $\epsilon$ and compare with the closed form (24): a mismatch in the $\epsilon\to0$ limit would locate the failure in the interchange step.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Basel problem reduces to a one-line differential equation. Because $\sum_{n\ge1}1/n^2=\frac43\sum_{n\ge1}1/(2n-1)^2=\frac13\sum_{n\ge1}1/(n-\frac12)^2$, the sum is $\frac13\psi_1(1/2)$, where $\psi_1$ is the derivative of the digamma function $\psi=\Gamma'/\Gamma$, and the trigamma reflection formula $\psi_1(x)+\psi_1(1-x)=\pi^2/\sin^2\pi x$ gives $\psi_1(1/2)=\pi^2/2$. Rather than importing the reflection formula, the paper derives a stronger statement: for $\varphi(x)=\psi(x)-\psi(1-x)$, two regularized integral representations, multiplied through the principal-value identity, yield $\varphi^2+\pi^2$ as the same integral that differentiation gives for $-\varphi'$; hence $\varphi'=\pi^2+\varphi^2$. The initial condition $\varphi(1/2)=0$ then forces $\varphi=-\pi\cot\pi x$, and differentiating this returns the trigamma reflection formula. The same mechanism, applied to higher derivatives, expresses $\zeta(2k)$ through tangent numbers and yields the recurrence $(k+\tfrac12)\zeta(2k)=\sum_{m=1}^{k-1}\zeta(2m)\zeta(2k-2m)$.

Load-bearing premise

The proof assumes that two slightly damped principal-value integrals can be multiplied together and their integration orders swapped before the damping is removed; the needed convergence justification is gestured at but not carried out, and if that interchange fails the differential equation behind the reflection formula is unsupported.

Editorial extensions

If this is right

The Basel identity $\sum_{n\ge1}1/n^2=\pi^2/6$ follows from the differential equation plus an elementary splitting into odd and even terms, with no sine product or Fourier series.
The same ODE yields the reflection formula for both the digamma and trigamma functions in one stroke.
Every even zeta value is expressible as $\zeta(2k)=\pi^{2k}T_{2k-1}/(2(2^{2k}-1)(2k-1)!)$, where $T_n$ are the tangent numbers.
The tangent numbers satisfy the recurrence $T_n=\sum_{r=0}^{n-1}\binom{n-1}{r}T_rT_{n-1-r}$, making $\zeta(2k)$ recursively computable.
The zeta values themselves obey $(k+\tfrac12)\zeta(2k)=\sum_{m=1}^{k-1}\zeta(2m)\zeta(2k-2m)$, a direct recurrence for the even zeta values.

Reading between the lines

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

The inverse-square potential is motivational scaffolding rather than a logical input: any regularization of $\sum 1/(n-x)$ that gives the digamma series would carry the same argument.
The step the author flags as informal, the interchange of integrations and the $\epsilon\to0$ limit in equations (22)-(24), is the natural place to supply a standard measure-theoretic justification; doing so would convert the derivation into a fully rigorous proof without changing the conclusion.
The same trick of multiplying two regularized principal-value integrals could be tried on other combinations of digamma-type integrals, potentially producing differential equations for other special functions that satisfy reflection-type identities.

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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 one thing to know: this is a well-written expository note that gives a genuinely fresh route to a very old result. The Coulomb-force analogy leads to the digamma function, and the key move is a self-contained proof of the reflection formula via the ODE φ′ = π² + φ², avoiding Euler's sine product. That derivation is elegant and, as far as I can tell, correct. The stress-test note is right: the passage through equations (21)–(24), where two Sokhotski–Plemelj regularized integrals are multiplied and the ε-limit is interchanged with integration, looks delicate but can be justified. For each ε the integrals are absolutely convergent, the substitution works, and the limit yields exactly what the author claims. So the central derivation stands.

What is actually new: not the result—ζ(2)=π²/6 and the even-zeta formulas are classical—but the presentation. The physical framing is simple and motivating, and the ODE proof of the reflection formula is a nice addition to the pedagogical literature. The extension to all ζ(2k) via tangent numbers and the recurrence for ζ(2k) are also cleanly derived, though the results themselves are known. The paper is honest about its own limitations: it acknowledges the informal limit interchanges and points to Fubini and dominated convergence, and it admits the physics is tenuous. No circularity: the trigamma value at 1/2 is derived from the reflection formula, which is proven independently.

Soft spots, in proportion: the ε-limit step is the only place I would want a referee to push for a bit more rigor—a short footnote or appendix justifying the interchange would remove the one lingering concern. The use of the Sokhotski–Plemelj formula is heavier than needed for an elementary proof, but it is standard and the author explains it. The proof of the digamma integral representation (16) is only sketched, but again it is standard. These are minor issues in an expository context, not flaws that undermine the argument.

The citation pattern is solid: the author engages with the vast literature, credits Wästlund's earlier physical approach, and gives historical references for Dedekind's proof. There is no fitting of parameters or invented entities.

My verdict: this paper deserves a serious referee. It is not a groundbreaking research contribution, but it is a clear, correct, and genuinely useful exposition that could well become a go-to reference for people who want a physicist-flavored proof of the Basel problem. I would send it to peer review for a teaching-oriented journal or as a math.HO note. If I were editing, I would ask for a small note on the ε-limit justification and then accept.

Referee Report

0 major / 4 minor

Summary. The paper presents a proof of Euler's Basel formula ζ(2)=π²/6 by first rewriting the series as one third of the trigamma value ψ₁(1/2), and then proving Euler's reflection formula for the digamma function through an elementary differential equation. The author defines φ(x)=ψ(x)−ψ(1−x), derives integral representations and a Sokhotski–Plemelj regularization, obtains the differential equation φ′(x)=π²+φ(x)², solves it, and differentiates to obtain ψ₁(1/2)=π²/2. The final sections extend the method to all even zeta values and derive a recurrence for ζ(2k) from the cotangent function.

Significance. The central derivation is sound and, in my reading, non-circular: the reduction of ζ(2) to ψ₁(1/2) does not presuppose the value of the series, and the proof of the reflection formula via the first-order ODE is independent of the target identity. The paper contains no parameter fitting, and the physical Coulomb analogy is explicitly declared to be motivational rather than load-bearing. The route through the differential equation is an attractive and reasonably elementary way to prove the reflection formula, and the generalization to all ζ(2k), with tangent numbers and the recurrence (45), is a nice bonus. The main delicate step, the ε→0 interchange in equations (21)–(24), is acknowledged by the author; I have checked that the step is justifiable, so I do not regard it as an error, though a brief verification would improve the exposition.

minor comments (4)

[Recurrence relation for ζ(2k), Eq. (37)] The recurrence T_n = Σ_{r=0}^{n-1} binom(n-1,r) T_r T_{n-1-r} is stated without qualification, but for n=1 it gives 0 rather than T_1=1; the recurrence is valid for n>1, because the constant term in 1+tan²x disappears only after a derivative is taken.
[Zeta function values at positive even integers, before Eq. (32)] The phrase 'differentiating (9) 2 k−) times' contains a typographical artifact and should read 'differentiating (9) 2k−1 times'.
[Proof of the reflection formula, Eqs. (21)–(24)] The Sokhotski–Plemelj formula is applied to the function t^{-x}, which is not a Schwartz test function; the footnote's smooth-test-function wording does not literally cover this case, so the step is formal as written. Since the conclusion is correct and can be justified by a cutoff argument followed by a limit, I suggest adding a sentence indicating that justification.
[Eq. (23)] The evaluation of the t-integral is stated without derivation; a brief indication using partial fractions and a limiting argument would help the reader verify the branch choice and the factor −ln y/(1−y).

Circularity Check

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No circularity found: the Basel sum is reduced to an independently derived trigamma reflection formula, not to the target identity.

full rationale

The derivation chain is self-contained: equation (2) is an exact parity rewrite of the series, and equation (8) identifies the Basel sum with (1/3)psi_1(1/2) via the standard trigamma series representation (30). The load-bearing step is the derivation of the digamma reflection formula, which is proved from the integral representation (19) and the Sokhotski-Plemelj computation leading to the differential equation (26). That computation, equations (21)-(25), does not assume the value of zeta(2) or any equivalent identity; it is an independent integral calculation. The initial condition phi(1/2)=0 follows from the definition (17), and solving (26) gives phi(x)=-pi cot(pi x), hence the trigamma reflection formula and psi_1(1/2)=pi^2/2. The physical Coulomb analogy in equations (3)-(4) is motivational and not load-bearing; removing it leaves the integral argument intact. There are no fitted parameters, no self-citations, and no known result smuggled in under a new name. The only delicate point is the interchange of limits in (21)-(24), which is a matter of analytic justification rather than circularity, since the identities are not being assumed from the target conclusion.

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

No free parameters are fitted. The proof uses standard gamma and digamma identities plus classical analytic justification; the Coulomb analogy introduces no new physical entity.

assumptions (4)

standard math Gamma function definition via Euler or Weierstrass product (Eqs. 10-11).
Used as the basis for the digamma identity (7) and the integral representation (16).
standard math Digamma integral representation: ψ(1-x) = -γ + ∫₀¹ (1-t^{-x})/(1-t) dt (Eq. 16).
Derived by geometric series expansion; the paper provides the derivation outline.
standard math Sokhotski-Plemelj formula: 1/(z ± iϵ) = P(1/z) ∓ iπδ(z) in the distributional sense (Eq. 20).
Used to convert principal-value integrals into limits of ordinary integrals.
standard math Fubini's theorem and dominated convergence justify interchanging integrals and limits in equations (22)-(24).
The paper explicitly acknowledges this need and cites reference [22], but does not verify the domination conditions in the text.

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Cite this review

Pith. "Pith review of Basel problem: a physicist's solution." pith.science (2026). https://pith.science/paper/NOIGNY4H

@misc{pith2026190807518,
  author       = {Pith},
  title        = {Pith review of: Basel problem: a physicist's solution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NOIGNY4H}},
  note         = {Machine review of arXiv:1908.07518}
}

read the original abstract

Some time ago Wastlund reformulated the Basel problem in terms of a physical system using the proportionality of the apparent brightness of a star to the inverse square of its distance. Inspired by this approach, we give another physical interpretation which, in our opinion, is simpler, natural enough, and very Eulerian in its spirit.

Discussion (0). Continue with ORCID to comment.

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] [1]

Feynman, R. (2017). The Character of Physical Law. Cambridge, MA: MIT Press, p. 168

work page 2017

[2] [2]

Moreno, S. (2016). A short and elementary proof of the Basel problem, College Math. J. 47(2):134-135. https://arxiv.org/abs/1502.07667

work page Pith review arXiv 2016

[3] [3]

Boyadzhiev, K.N. (2015). Some integrals related to the Basel pr oblem, Ci- entia. A 26:1-13

work page 2015

[4] [4]

Glebov, G. (2015). A peculiar proof of an identity of Euler. Math. Gaz. 99(544):139-143

work page 2015

[5] [5]

Ghorbanpour, A., Hatzel, M. (2017). Parseval’s Identity and Va lues of Zeta Function at Even Integers, https://arxiv.org/abs/1709.09326

arXiv 2017

[6] [6]

Costandin, M. (2017). A new proof for the exact values of ζ(2k) for k ∈ N, https://arxiv.org/abs/1712.02255

work page Pith review arXiv 2017

[7] [7]

Siklos, S. (2018). A method of evaluating ζ(2) and ∞∫ 0 sin x x dx, Math. Gaz. 102(553):114-121

work page 2018

[8] [8]

Shenvi Pause (2018)

Kapil R. Shenvi Pause (2018). Basel Problem: A Solution Motivat ed by the Power of a Point, Amer. Math. Monthly 125(6): 558-560

work page 2018