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REVIEW 1 major objections 7 minor 7 references

Ramanujan, Landau and Casimir, divergent series: a physicist point of view

T0 review · 1 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that Ramanujan's −1/12 is not a formal curiosity but a real constant that appears in measurable physical quantities such as the Landau susceptibility and the Casimir force.

desk verdict A genuinely useful teaching paper on zeta regularization that deserves refereeing, but the 3D Casimir pressure is off by a factor of 3 and the smoothing-dependence caveat needs more care. read the letter →

arxiv 2506.08664 v1 pith:KNFV5DDW submitted 2025-06-10 cond-mat.other cond-mat.mes-hall

Gilles Montambaux This is my paper
classification cond-mat.othercond-mat.mes-hall
keywords RamanujansummationRiemannzetafunctionregularizationLandaudiamagnetismCasimireffectPoissonformuladivergentseriesdeHaas–vanAlphenoscillations
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

The paper sets out to give the notorious result $1+2+3+\cdots=-1/12$ a physical meaning rather than treating it as a formal trick. It proposes a systematic extraction rule: replace a divergent sum $\sum_n n^p$ by the step function $S_p(x)=\sum_{n\ge1} n^p\,\Theta(x-n)$, decompose this function by Poisson summation, and discard the oscillatory terms; what remains is a power-law term $x^{p+1}/(p+1)$ plus a constant $\zeta(-p)$, which the paper identifies with the Ramanujan sum. The claim is that this constant appears in measurable physics: in the Landau diamagnetic susceptibility of a two-dimensional electron gas (from the sum over odd integers, with $\zeta(-1)=1/12$) and in the Casimir pressure (from the sum over cubes, with $\zeta(-3)=1/120$). A sympathetic reader should care because the paper turns a famous paradox into a specific, testable statement about which constants of nature encode the regularized value of a divergent series.

What carries the argument

The machine that carries the argument is the Poisson summation formula applied to the step function $S_p(x)=\sum_{n\ge1} n^p\,\Theta(x-n)$. Writing the sum as a Fourier series separates it into three pieces: the smooth average $x^{p+1}/(p+1)$, a set of periodic terms that oscillate with zero mean, and the constant $2\Gamma(p+1)/(2\pi)^{p+1}\,\cos((p+1)\pi/2)\,\zeta(p+1)$, which the Riemann functional equation rewrites as $\zeta(-p)$. A second piece of machinery is the smoothing prescription: replacing the sharp Heaviside cutoff by a smooth function $\chi_\beta(\epsilon_F-\epsilon)$ whose width $1/\beta$ is much smaller than $\epsilon_F$ damps the oscillations through a reduction factor (Fermi-Dirac gives $\pi Z/\sinh(\pi Z)$; impurity scattering gives $e^{-Z/2}$) without changing the power-law or constant contributions. The physical content is that the Landau-level sum and the Casimir mode sum are exactly such step functions, so the measured constants are the Ramanujan constants.

What would settle it

Compute the constant term in the Poisson decomposition of $S_p(x)$ using a Gaussian or other smooth cutoff and check numerically whether it converges to $\zeta(-p)$; the paper's own Appendix D shows an exponential cutoff gives a different power-law coefficient, so any smooth cutoff that also changes the constant would refute the claimed universality. An experimental falsifier would be a high-precision measurement of the 2D orbital susceptibility or the Casimir pressure that disagrees with the coefficients $-e^2/(24\pi m_e)$ or $-\pi^2\hbar c/(720a^4)$ after all oscillations are damped.

Watch

Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is that for every integer power $p$ the partial-sum step function $S_p(x)$ obeys $S_p(x)=x^{p+1}/(p+1)+\zeta(-p)+\mathrm{osc}(x)$, where $\mathrm{osc}(x)$ is an oscillatory function whose average is zero. The constant $\zeta(-p)$ is the Ramanujan sum of the divergent series $\sum_{n=1}^{\infty} n^p$, obtained here by Fourier analysis of the step function and removal of harmonics. In the two physical examples the constant is not a curiosity: after the oscillations are damped by temperature or impurity scattering, the energy of a two-dimensional electron gas in a magnetic field contains a term proportional to $1/24$, which yields the Landau susceptibility $-e^2/(24\pi m_e)$; similarly, the zero-point energy between two plates contains a term proportional to $\zeta(-3)=1/120$ that yields the Casimir pressure $-\pi^2\hbar c/(720 a^4)$. The paper stresses that the divergent power-law part is a vacuum (Fermi-sea) energy, while the Ramanujan constant is the part revealed by an external parameter such as the magnetic field or the plate separation.

Load-bearing premise

The argument relies on the assumption that replacing the sharp cutoff by a smooth cutoff of width much smaller than the running variable removes the oscillatory terms while leaving the power-law term and the constant term unchanged; the paper checks this only for Fermi-Dirac and impurity-broadened cutoffs and shows in Appendix D that an exponential cutoff changes the power-law coefficient, so the invariance is not established for arbitrary smoothing.

Editorial extensions

If this is right

  • The Landau susceptibility of a 2D electron gas, $-e^2/(24\pi m_e)$, is a direct experimental window into the Ramanujan value $1/12$ of the divergent sum of odd integers.
  • The 1D Casimir force, $-\pi\hbar c/(24a^2)$, is a direct experimental window into $\zeta(-1)=-1/12$; the 3D Casimir pressure, $-\pi^2\hbar c/(720a^4)$, is a window into $\zeta(-3)=1/120$.
  • Because $S_p(x)=x^{p+1}/(p+1)+\zeta(-p)+\mathrm{osc}(x)$ holds for every integer $p$, the paper implies that the Ramanujan value of any power sum is the zeta value at a negative integer, with even-power sums vanishing.
  • The same Poisson-based recipe assigns well-defined values to other divergent or conditionally convergent series: the sum of ones is $-1/2$, the alternating sum of integers is $1/4$, the Grandi series is $1/2$, and the harmonic series has the Euler constant as its constant term.
  • Finite temperature or impurity scattering damps the oscillations without changing the physically relevant constant, which explains why the measured susceptibility and Casimir force are independent of those damping mechanisms.

Reading between the lines

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

  • A testable extension suggested by the method is to search for other measurable responses, for instance in higher-dimensional or curved geometries, where the mode sums involve powers other than 1 and 3 and would reveal other zeta values $\zeta(-p)$.
  • The paper's Appendix D shows that not every smooth cutoff preserves the extraction: an exponential cutoff changes the power-law coefficient. This suggests that a full statement of the physicist's regularization needs a precise class of admissible cutoffs, and the physical argument itself supplies the criterion: the cutoff must be a smoothed Heaviside step whose width is much smaller than the runni
  • Read as a pedagogical bridge, the step-function decomposition offers a way to present zeta-regularized sums to students without invoking analytic continuation, framing the constant as the zero-frequency Fourier component of the partial-sum staircase.
  • The analogy between the Landau and Casimir calculations hints at a general dictionary: the divergent part of a mode sum is the vacuum energy, and the measurable response to an external parameter is the Ramanujan constant; if this dictionary holds, it could guide intuition for when regularization artifacts contaminate predictions in other field-theoretic settings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. The paper proposes a pedagogical method for making sense of Ramanujan-type divergent sums such as 1+2+3+... = -1/12. The method consists of representing a partial sum S_p(x) = Σ n^p Θ(x-n) as a step function, applying Poisson summation, and identifying the non-oscillating constant term in the resulting decomposition with the Ramanujan sum. The paper derives the general result S_p(x) = x^{p+1}/(p+1) + ζ(-p) + osc(x), then applies it to the Landau diamagnetism of a 2D electron gas and to the Casimir effect in 1D and 3D, arguing that the field-dependent or distance-dependent constant terms in these physical quantities are physical manifestations of Ramanujan sums.

Significance. If the central claims are taken at face value, the paper gives a clean, self-contained derivation of a standard but often mystifying result, and it ties the zeta-regularized constants to concrete physical observables. The Poisson-summation route and the use of the Riemann functional equation are correct and parameter-free; the paper is honest about a counterexample to naive regularization (Appendix D) and about the physical subtleties of the vacuum energy. The pedagogical value is real, though no new mathematical result is claimed. However, the 3D Casimir pressure quoted in Eq. (64) is inconsistent with the paper's own energy expression Eq. (62) by a factor of 3, so the quantitative content of one of the two advertised physical manifestations is wrong as stated and must be corrected before the manuscript is suitable for publication.

major comments (1)
  1. [Sec. 7.3, Eq. (64)] The Casimir pressure is obtained from the Casimir energy E_C(a) = -L²π²ħc/(720a³) in Eq. (62) by P = -(1/L²) ∂E_C/∂a. Since E_C ∝ a^{-3}, differentiation gives a factor 3, yielding P = -π²ħc/(240a⁴), which is the standard result for two perfectly conducting plates. As written, Eq. (64) instead gives -π²ħc/(720a⁴), missing this factor. This is not a cosmetic slip: Eq. (64) is the advertised physical realization of ζ(-3), and the numerical constant is the quantitative content of that identification. The conceptual connection survives, but the manuscript's central physical example contains an incorrect constant until Eq. (64) is corrected.
minor comments (7)
  1. [Appendix B, Eq. (69)] The Poisson formula for S0(x) is quoted with a plus sign before 1/2, but the correct non-oscillating constant is -1/2, as stated in Eq. (36). The sign error does not affect the main results because Eq. (60) in effect uses the correct S0(x) = x - 1/2 + osc(x), which is needed to obtain the cancellation in Eq. (61).
  2. [Sec. 2, Eq. (17) and Appendix C, Eq. (86)] The definition of Z_m in Eq. (17), Z_m = 2mπ²/(βω_c), is inconsistent with the Z used in Eq. (85)-(86), where Z = 2πm/(βω_c). With the latter, R(Z) = πZ/sinh(πZ) reproduces the standard thermal damping factor; with the former, an extra π appears. Please reconcile the two definitions.
  3. [Conclusion, final equations] The concluding display writes 1+3+5+...+x → x²/2 + 1/12, but Eq. (18) states the same sum scales as x²/4 + 1/12. The x²/2 scaling applies to the sum of half-integers in the Landau problem; please correct the conclusion or explicitly state which convention is being used.
  4. [Sec. 6.1 and Appendix C] In Eq. (36), the summation index starts at m=0 but the sine term vanishes for m=0; it should start at m=1. Also, the phrase in Appendix C 'with a scale larger much smaller than x' is garbled; it should say 'with a transition width much larger than the level spacing and much smaller than x.'
  5. [Appendix C, final sentence] The claim that a smooth cutoff 'kills the oscillations but does not modify the infinite and the Ramanujan sum' is too broad, as the paper itself shows in Appendix D that an exponential cutoff modifies the leading power law. Please state explicitly the class of smoothed step functions for which the Poisson constant is invariant, or reformulate the claim as a condition on the smoothing kernel rather than a general statement.
  6. [Appendix B, Eq. (73)] The last term in the integral formula for ∫_0^x (x³-y³) cos(ωy) dy is written as 6ω sin(2ωx); by dimensional consistency this should presumably involve x, e.g., 6ωx sin(ωx). Please verify and correct the formula, even though it is not used in the main text.
  7. [References] Reference [1] is a pair of Wikipedia URLs; for a journal article, a conventional reference such as G.H. Hardy's 'Divergent Series' would be more appropriate and more stable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Ramanujan constants are derived from Poisson summation and the Riemann functional equation, not assumed as inputs.

full rationale

The paper's central derivation is self-contained and uses standard external mathematics. The Ramanujan constant for each divergent series is obtained by applying the Poisson summation formula to a step function (Appendix B), isolating the non-oscillating constant, and then using the Riemann functional equation to express that constant as ζ(−p). For example, Eq. (21) gives S(x)=x²/2+osc, and the constant −1/12 follows from the convergent Basel sum Σ1/m²=π²/6; Eq. (75) generalizes this to Sp(x)=x^{p+1}/(p+1)+ζ(−p). No parameter is fitted to the targeted physical result, and no load-bearing self-citation appears. The Landau susceptibility (Eq. 13) and the 1D Casimir force (Eq. 55) are derived from the same Poisson machinery, and the identification with ζ(−1) and ζ(−3) is made after the constants are computed, not before. The paper also explicitly checks in Appendix D that the constant term is robust to replacing the Heaviside cutoff with an exponential one, while the power-law coefficient changes—showing awareness that the extraction procedure is a definition and testing its stability. The 3D Casimir pressure in Eq. (64) contains an arithmetic factor error (differentiating Eq. (62) gives −π²ħc/(240a⁴), not −π²ħc/(720a⁴)), but that is a correctness defect, not a circularity, because the conceptual derivation of the ζ(−3) contribution does not depend on that slip. Overall, the claim that the Ramanujan sum appears in physical quantities is a genuine derivation, not a restatement of its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters and no new entities. The paper relies on standard mathematical identities, standard physical cutoffs, and an assumed smoothing invariance; these are listed above.

assumptions (5)
  • standard math The Poisson summation formula (Eq. 68) can be applied to f(y) = y^p Theta(x-y) with a discontinuous Heaviside factor.
    Used in Appendix B to derive S_p(x); a rigorous treatment would justify the distributional evaluation at the discontinuity, which the paper does not provide.
  • standard math The Riemann functional equation (Eq. 95) is valid and can be used to replace sums over m with zeta(1-p).
    Invoked after Eq. (74) to obtain S_p(x) = x^(p+1)/(p+1) + zeta(-p).
  • domain assumption The Fermi energy epsilon_F (Landau) and plasma frequency omega_p (Casimir) act as sharp physical cutoffs, and smoothing with chi_beta suppresses oscillations without changing the power-law and constant terms.
    Central to the method; introduced in Sections 2 and 7, examined in Appendices C and D, where the exponential cutoff shows the caveat.
  • domain assumption The vacuum-energy term L^2 a hbar omega_p^4 / (8 pi^2 c^3) in the Casimir energy is the same inside and outside the plates and therefore contributes no force.
    Stated in Section 7.3 and footnote 5; necessary to obtain the Casimir pressure from the total energy.
  • domain assumption In the Landau calculation the zero-field energy m_e epsilon_F^2 / (4 pi hbar^2) is the reference energy; only the B-dependent term is the physical diamagnetic response.
    Used in Section 2 to identify the Landau susceptibility from the field-dependent constant term.

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Cite this review

Pith. "Pith review of Ramanujan, Landau and Casimir, divergent series: a physicist point of view." pith.science (2026). https://pith.science/paper/KNFV5DDW

@misc{pith2026250608664,
  author       = {Pith},
  title        = {Pith review of: Ramanujan, Landau and Casimir, divergent series: a physicist point of view},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNFV5DDW}},
  note         = {Machine review of arXiv:2506.08664}
}
read the original abstract

It is a popular paradoxical exercise to show that the infinite sum of positive integer numbers is equal to -1/12, sometimes called the Ramanujan sum. Here we propose a qualitative approach, much like that of a physicist, to show how the value -1/12 can make sense and, in fact, appears in certain physical quantities where this type of summation is involved. At the light of two physical examples, taken respectively from condensed matter -- the Landau diamagnetism -- and quantum electrodynamics -- the Casimir effect -- that illustrate this strange sum, we present a systematic way to extract this Ramanujan term from the infinity.

Figures

Figures reproduced from arXiv: 2506.08664 by the authors.

Figure 1
Figure 1. Density of states of 2D free electrons. Blue: zero field. In a magnetic field, the quantum states con￾dense into Landau levels, of energy (n + 1/2)ℏωc and de￾generacy eB/h. We consider the total energy of a two-dimensional (2D) electron gas. Due to Fermi principle, the elec￾tron states are populated up to an energy called the Fermi energy ϵF which depends on the electron den￾sity. Figure (1) recalls how the electron… view at source ↗
Figure 2
Figure 2. Total energy of 2D electron gas in a mag￾netic field, as function of the ratio ϵF /ℏωc. T = 0K: black steps. Finite temperature kBT ≪ ℏωc, blue curve. At temperature T such that ℏωc ≪ kBT ≪ ϵF , one re￾covers the zero field quadratic increase, with a small field dependent correction, not visible at this scale (red curve). is the zero field total energy. b) The oscillatory terms are signatures of the step-like form o… view at source ↗
Figure 3
Figure 3. Red dots: discrete values s(n). Green: the step function S(x). Black: the functions x(x + 1)/2 and x(x − 1)/2. Blue: the function x 2 /2 − 1/12. We now come to our original problem. Consider the sum of natural numbers. The same procedure as 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Fit of the function S(x) by the Fourier series, limited here to 5 terms, then to a single term, and then with no harmonics. In black, without the factor −1/12; in blue, with the factor −1/12 (the difference is barely visible). which is shown on Fig.4. We have learned i…
Figure 5
Figure 5. Figure 5: Geometry of the Casimir effect. Two metallic plates at short distances are attracted by a small force. 7.2 Casimir effect in one dimension It is instructive to start with a 1D calculation, since it is very similar to the Landau calculation. We con￾sider the space confi…
Figure 6
Figure 6. Figure 6: Top: different cut-off functions. Θ(x − n) (red) , χβ(x−n) (blue), χτ (x−n) (purple), e −n/x (black). Here we have taken n = 1, β = τ = 10. Bottom: sum of integers with the corresponding cut-off function. Obvi￾ously the exponential cut-off is not appropriate. step-like…

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    https://en.wikipedia.org/wiki/1_\ https://en.wikipedia.org/wiki/Ramanujan_summation

    work page

  2. [2]

    L. D. Landau, Diamagnetism of metals , Z. Phys., 64 , 629 (1930); On the de Haas-van Alphen effect , Proc. Roy. Soc., A170 , 383 (1939); in Collected Papers od L. D. Landau, D. Ter Haar ed., Gordon and Breach (1965)

    work page 1930

  3. [3]

    Shoenberg, Magnetic oscillations in metals , Cambridge University Press (1984)

    D. Shoenberg, Magnetic oscillations in metals , Cambridge University Press (1984)

    work page 1984

  4. [4]

    H. B. G. Casimir, On the Attraction Between Two Perfectly Conducting Plates , Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 51 , 793 (1948)

    work page 1948

  5. [5]

    Duplantier, Introduction \`a l'effet Casimir , S\'eminaire Poincar\'e, 1 , 21 (2002)

    B. Duplantier, Introduction \`a l'effet Casimir , S\'eminaire Poincar\'e, 1 , 21 (2002)

    work page 2002

  6. [6]

    Riemann functional relation : 2 (2 )^s [s] s 2 (s)= (1-s)

    work page

  7. [7]

    !1A Qa

    https://en.wikipedia.org/wiki/1_ https://en.wikipedia.org/wiki/Ramanujan_summation spectre-landau.jpg0000664000000000000000000046077514712536340013217 0ustar rootroot JFIF J JC C | " !1A Qa "q 2 #B R 3br w !1 AQ aq "2 B #3R br 4 Jaq#\ 3 Oݿ | 8 X<BM[>Lea;N aڼ C E?uE[wQ\ cT8=N*. ( ( ( _ ࿍/|8ῌ/ 'S Ɇ H' 7b ^E|m ﮾ |H_ |_qe*yIy|;F0KG / M vQ?e[KO M3 dUO& #eY"'g9...

    work page

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Reviewed August 7, 2026 · model on record in the stance chip above.