pith. sign in

arxiv: 2606.09977 · v1 · pith:C5PZ5NTFnew · submitted 2026-06-08 · ✦ hep-th · math-ph· math.MP

Zeta Functions and the Superstring

Grant N. Remmen This is my paper

Pith reviewed 2026-06-27 15:34 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords superstring amplitudeMellin transformRiemann zeta functiondispersion relationeffective field theory expansionforward limitmomentum transfer
0
0 comments X

The pith

The Mellin transform of the superstring amplitude reduces exactly to the Riemann zeta function in the forward limit.

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

The paper computes the Mellin transform of the superstring amplitude with respect to energy at fixed momentum transfer. In the forward limit this transformed object equals the Riemann zeta function, while at nonzero t it forms a deformation of zeta that still inherits properties from the string amplitude. Using dispersion relations with integer subtractions, the transform supplies a closed-form expression for the effective field theory expansion of the amplitude order by order in s at finite t.

Core claim

The Mellin transform in energy of the superstring amplitude at fixed t reduces exactly to the Riemann zeta function when t equals zero. For t not equal to zero the object is a deformation of zeta. Physical features of the string amplitude produce the exact reduction and the other mathematical properties, and the associated dispersion relation with integer subtractions furnishes a new closed-form expression for the EFT expansion of the amplitude order by order in s at finite t.

What carries the argument

The Mellin transform of the superstring amplitude, which converts the amplitude into a zeta-like function whose properties follow from the amplitude's physical attributes.

If this is right

  • The dispersion relation defined by the Mellin transform supplies the EFT expansion coefficients order by order in s at any fixed t.
  • Integer subtractions in the dispersion relation produce the closed-form expressions for each order.
  • At nonzero t the transformed amplitude is a continuous deformation of the zeta function that retains the same physical origin.
  • The reduction to zeta occurs precisely when momentum transfer vanishes.

Where Pith is reading between the lines

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

  • The appearance of zeta may indicate that number-theoretic objects organize the low-energy expansion of string amplitudes more generally.
  • Similar Mellin transforms could be applied to other string or field-theory amplitudes to search for analogous closed forms.
  • The deformation at finite t might be studied as a function of t to reveal how the zeta reduction is lost away from the forward limit.

Load-bearing premise

The physical attributes of the string amplitude are what produce the exact reduction of its Mellin transform to the zeta function in the forward limit.

What would settle it

Direct numerical evaluation of the Mellin transform of the superstring amplitude in the forward limit at several positive real values, checking whether the results match the corresponding values of the Riemann zeta function.

Figures

Figures reproduced from arXiv: 2606.09977 by Grant N. Remmen.

Figure 1
Figure 1. Figure 1: Illustration of the contour integral defining [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Zeros of Ω(z, t) in the complex z plane for integer t. Only zeros off of the real axis are plotted. Formulas are given in Eqs. (13) and (15) for the positive and negative cases, respectively. Left: Positive case, where t = k for k = 0 (green) through k = 30 (purple). For the cases k = 0 and 1, Ω(z, k) corresponds to the Riemann zeta function as in Eq. (9). Right: Negative case, where t = k for k = −10 (red… view at source ↗
read the original abstract

The superstring amplitude's Mellin transform in energy is computed at fixed momentum transfer. In the forward limit, it is shown that this transformed amplitude reduces to the Riemann zeta function, while for $t\neq 0$ it represents a deformation of $\zeta$. This object exhibits remarkable mathematical properties as a result of physical attributes of the string amplitude. For integer subtractions, the dispersion relation defined by the Mellin transform yields the effective field theory expansion of the string amplitude order by order in $s$ at finite $t$, for which a new closed-form expression is derived.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript computes the Mellin transform (in the energy variable, at fixed momentum transfer t) of the tree-level superstring amplitude. It asserts that this object reduces exactly to the Riemann zeta function in the forward limit t→0, represents a deformation of zeta for t≠0, and that the associated dispersion relation with integer subtractions supplies a new closed-form expression for the EFT expansion of the amplitude order-by-order in s at finite t.

Significance. If the central reduction and closed-form derivation hold with the stated independence from auxiliary choices, the result would furnish an explicit mathematical bridge between the gamma-function structure of superstring amplitudes and zeta functions, potentially yielding parameter-free expressions for the low-energy expansion that could be checked against known string EFT coefficients.

major comments (1)
  1. [Abstract] Abstract: the central claim that the Mellin transform reduces exactly to the Riemann zeta function in the forward limit is stated without any definition of the transform (normalization, integration contour, principal-value prescription, or handling of the simple poles of the gamma-function amplitude at positive integers in s). This specification is load-bearing for the subsequent assertion that the dispersion relation yields a new closed-form EFT expansion; without it, one cannot determine whether the reduction is independent or follows by construction from the chosen regularization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater precision in the abstract. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the Mellin transform reduces exactly to the Riemann zeta function in the forward limit is stated without any definition of the transform (normalization, integration contour, principal-value prescription, or handling of the simple poles of the gamma-function amplitude at positive integers in s). This specification is load-bearing for the subsequent assertion that the dispersion relation yields a new closed-form EFT expansion; without it, one cannot determine whether the reduction is independent or follows by construction from the chosen regularization.

    Authors: We agree that the abstract, as currently written, does not specify the definition of the Mellin transform. The full manuscript defines the transform in Section 2 as the integral ∫ ds s^{z-1} A(s,t) along a vertical contour in the complex s-plane (Re(s) > 0, indented with a principal-value prescription to avoid the simple poles of the gamma-function factors at positive integers), with normalization chosen so that the forward limit t → 0 recovers the standard Mellin representation of the zeta function. The reduction is derived after this definition and is shown to be independent of the precise indentation provided the contour remains to the right of all poles for the subtracted dispersion relation. We will revise the abstract to include a concise statement of this definition, contour choice, and principal-value handling so that the central claim is self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation chain not reducible to inputs from provided text

full rationale

The abstract asserts that the Mellin transform of the superstring amplitude reduces to the Riemann zeta function in the forward limit and yields a closed-form EFT expansion via dispersion relations for integer subtractions. No equations, definitions of the Mellin contour, pole prescriptions, or derivation steps are available in the given material to inspect for self-definitional equivalence, fitted parameters renamed as predictions, or load-bearing self-citations. The claim is presented as following from physical attributes of the amplitude, with no exhibited reduction to its own inputs by construction. The paper is therefore scored as self-contained with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all entries are therefore empty.

pith-pipeline@v0.9.1-grok · 5609 in / 1051 out tokens · 27639 ms · 2026-06-27T15:34:40.128285+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 9 linked inside Pith

  1. [1]

    The pair correlation of zeros of the zeta function,

    H. Montgomery, “The pair correlation of zeros of the zeta function,” Analytic number theory, Proc. Sympos. Pure Math.24(1973) 181

    1973

  2. [2]

    Pólya, c

    G. Pólya, c. 1913. Unpublished. See A. Odlyzko,Cor- respondence about the origins of the Hilbert-Pólya Con- jecture, Univ. Minnesota

    1913

  3. [3]

    H=xpand the Rie- mann Zeros,

    M. V. Berry and J. P. Keating, “H=xpand the Rie- mann Zeros,” inSupersymmetry and Trace Formulae: Chaos and Disorder, I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii, eds., vol. 370 ofNATO ASI Series (Series B: Physics). Springer, Boston, 1999

    1999

  4. [4]

    Nonclassical Degrees of Freedom in the Riemann Hamiltonian,

    M. Srednicki, “Nonclassical Degrees of Freedom in the Riemann Hamiltonian,”Phys. Rev. Lett.107(2011) 100201,arXiv:1105.2342 [math-ph]

    Pith/arXiv arXiv 2011

  5. [5]

    Hamilto- nian for the zeros of the Riemann zeta function,

    C.M.Bender, D.C.Brody, andM.P.Müller, “Hamilto- nian for the zeros of the Riemann zeta function,”Phys. Rev. Lett.118(2017) 130201,arXiv:1608.03679 [quant- ph]

    Pith/arXiv arXiv 2017

  6. [6]

    Stochasticbehaviorinquantumscat- tering,

    M.C.Gutzwiller, “Stochasticbehaviorinquantumscat- tering,”Physica D: Nonlinear Phenomena7(1983) 341

    1983

  7. [7]

    The phase of the Riemann zeta function and the inverted harmonic oscillator,

    R. K. Bhaduri, A. Khare, and J. Law, “The phase of the Riemann zeta function and the inverted harmonic oscillator,”Phys. Rev. E52(1995) 486,arXiv:chao- dyn/9406006

    arXiv 1995

  8. [8]

    Generalized Riemann hypothesis, time series and normal distributions,

    A. LeClair and G. Mussardo, “Generalized Riemann hypothesis, time series and normal distributions,”J. 6 Stat. Mech.1902(2019) 023203,arXiv:1809.06158 [math.NT]

    Pith/arXiv arXiv 1902

  9. [9]

    Riemann zeros as quan- tized energies of scattering with impurities,

    A. LeClair and G. Mussardo, “Riemann zeros as quan- tized energies of scattering with impurities,”JHEP04 (2024) 062,arXiv:2307.01254 [hep-th]

    arXiv 2024

  10. [10]

    Amplitudes and the Riemann Zeta Function,

    G. N. Remmen, “Amplitudes and the Riemann Zeta Function,”Phys. Rev. Lett.127(2021) 241602, arXiv:2108.07820 [hep-th]

    arXiv 2021

  11. [11]

    Adelic string ampli- tudes,

    P. G. O. Freund and E. Witten, “Adelic string ampli- tudes,”Phys. Lett. B199(1987) 191

    1987

  12. [12]

    From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis,

    Y.-H. He, V. Jejjala, and D. Minic, “From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis,”Int. J. Mod. Phys. A31(2016) 1650201, arXiv:1501.01975 [hep-th]

    Pith/arXiv arXiv 2016

  13. [13]

    Genus-zero and genus-one string amplitudes and special multiple zeta values,

    D. Zagier and F. Zerbini, “Genus-zero and genus-one string amplitudes and special multiple zeta values,” Commun. Number Theory Phys.14,arXiv:1906.12339 [math.NT]

    arXiv 1906

  14. [14]

    Motivic Multiple Zeta Values and Superstring Amplitudes,

    O. Schlotterer and S. Stieberger, “Motivic Multiple Zeta Values and Superstring Amplitudes,”J. Phys. A46 (2013) 475401,arXiv:1205.1516 [hep-th]

    Pith/arXiv arXiv 2013

  15. [15]

    Superstring amplitudes, uni- tarily, andHankeldeterminantsofmultiplezetavalues,

    M. B. Green and C. Wen, “Superstring amplitudes, uni- tarily, andHankeldeterminantsofmultiplezetavalues,” JHEP11(2019) 079,arXiv:1908.08426 [hep-th]

    arXiv 2019

  16. [16]

    Specifically, we are considering the polarization- stripped amplitude of the open superstring [46, 47], which is similar to the Veneziano amplitude of the bosonic string [48], with its characteristic Euler beta functionform.Ofcourse, theKoba-Nielsenintegralform of the string amplitude [49, 50] is a Mellin transform on the worldsheet variables, but here we...

  17. [17]

    Celestial amplitudes from UV to IR,

    N. Arkani-Hamed, M. Pate, A.-M. Raclariu, and A. Strominger, “Celestial amplitudes from UV to IR,” JHEP08(2021) 062,arXiv:2012.04208 [hep-th]

    arXiv 2021

  18. [18]

    The stringy S-matrix bootstrap: maximal spin and superpolynomial soft- ness,

    K. Häring and A. Zhiboedov, “The stringy S-matrix bootstrap: maximal spin and superpolynomial soft- ness,”JHEP10(2024) 075,arXiv:2311.13631 [hep-th]

    arXiv 2024

  19. [19]

    Strings from Almost Nothing,

    C. Cheung, G. N. Remmen, F. Sciotti, and M. Tarquini, “Strings from Almost Nothing,”arXiv:2508.09246 [hep- th]

    Pith/arXiv arXiv

  20. [20]

    These two forms ofζ(z)are manifestly the same using the behavior of gamma functions under shifts

    In other words, we have constructed two interesting in- tegral forms of the Riemann zeta function, expressed via the definition ofΩ(z,t), ζ(z) =−lim t→0 1 2πi ˛ C ds sz−1 Γ(−s)Γ(−t) Γ(1−s−t) =−lim t→1 1 2πi ˛ C ds sz Γ(−s)Γ(−t) Γ(1−s−t). These two forms ofζ(z)are manifestly the same using the behavior of gamma functions under shifts

  21. [21]

    Stringy dynamics from an amplitudes bootstrap,

    C. Cheung and G. N. Remmen, “Stringy dynamics from an amplitudes bootstrap,”Phys. Rev. D108(2023) 026011,arXiv:2302.12263 [hep-th]

    arXiv 2023

  22. [22]

    Bootstrap Principle for the Spectrum and Scattering of Strings,

    C. Cheung, A. Hillman, and G. N. Remmen, “Bootstrap Principle for the Spectrum and Scattering of Strings,” Phys. Rev. Lett.133(2024) 251601,arXiv:2406.02665 [hep-th]

    arXiv 2024

  23. [23]

    Causality, analyticity and an IR obstruction to UV completion,

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nico- lis, and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,”JHEP10(2006) 014, arXiv:hep-th/0602178

    Pith/arXiv arXiv 2006

  24. [24]

    Multipositivity bounds for scattering amplitudes,

    C. Cheung and G. N. Remmen, “Multipositivity bounds for scattering amplitudes,”Phys. Rev. D112(2025) 016017,arXiv:2505.05553 [hep-th]

    arXiv 2025

  25. [25]

    Energy’s and amplitudes’ positivity,

    A. Nicolis, R. Rattazzi, and E. Trincherini, “Energy’s and amplitudes’ positivity,”JHEP05(2010) 095, arXiv:0912.4258 [hep-th]. [Erratum:JHEP11(2011) 128]

    Pith/arXiv arXiv 2010

  26. [26]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables, vol. 55 ofApplied Mathematics Series. National Bureau of Standards, 1964. See Table 24.2

    1964

  27. [27]

    Analytic con- tinuation of multiple zeta-functions and their values at non-positive integers,

    Y. T. Shigeki Akiyama, Shigeki Egami, “Analytic con- tinuation of multiple zeta-functions and their values at non-positive integers,”Acta Arith.98(2001) 107

    2001

  28. [28]

    Analytic properties of multiple zeta- functions in several variables,

    K. Matsumoto, “Analytic properties of multiple zeta- functions in several variables,” inNumber Theory, W. Zhang and Y. Tanigawa, eds., p. 153. Springer US, Boston, MA, 2006

    2006

  29. [29]

    Analytic continuation of multiple zeta func- tions,

    J. Zhao, “Analytic continuation of multiple zeta func- tions,”Proc. Am. Math. Soc.128(2000) 1275

    2000

  30. [30]

    Global series for height1multiple zeta functions,

    P. T. Young, “Global series for height1multiple zeta functions,”Euro. J. Math.9(2023) 99

    2023

  31. [31]

    Series of height one multiple zeta func- tions,

    P. T. Young, “Series of height one multiple zeta func- tions,”Integers24(2024) A43

    2024

  32. [32]

    Multiple zeta values of fixed weight, depth, and height,

    Y. Ohno and D. Zagier, “Multiple zeta values of fixed weight, depth, and height,”Indag. Mathem.12(2001) 483

    2001

  33. [33]

    Multiple zeta values: Tasting notes,

    W. Zudilin, “Multiple zeta values: Tasting notes,” 2025. Unpublished notes

    2025

  34. [34]

    On multiple zeta values of extremal height,

    M. Kaneko and M. Sakata, “On multiple zeta values of extremal height,”Bull. Aust. Math. Soc.93(2016) 186

    2016

  35. [35]

    Simple Zeros of the Riemann Zeta-Function,

    J. Conrey, A. Ghosh, and S. Gonek, “Simple Zeros of the Riemann Zeta-Function,”Proc. London Math. Soc. 76(1998) 497

    1998

  36. [36]

    Note that throughout this section we have taken(s,t) to be in a region where the sums or integrals converge, and then tacitly analytically continued to all(s,t)

  37. [37]

    Uniqueness of the Veneziano representa- tion,

    D. D. Coon, “Uniqueness of the Veneziano representa- tion,”Phys. Lett. B29(1969) 669

    1969

  38. [38]

    Accumulation-point amplitudes in string theory,

    J. Maldacena and G. N. Remmen, “Accumulation-point amplitudes in string theory,”JHEP08(2022) 152, arXiv:2207.06426 [hep-th]

    arXiv 2022

  39. [39]

    Factorization and the generalized Veneziano model with satellites,

    D. J. Gross, “Factorization and the generalized Veneziano model with satellites,”Nucl. Phys. B13 (1969) 467

    1969

  40. [40]

    Bespoke dual res- onance,

    C. Cheung and G. N. Remmen, “Bespoke dual res- onance,”Phys. Rev. D108no. 8, (2023) 086009, arXiv:2308.03833 [hep-th]

    arXiv 2023

  41. [41]

    Multiparticle Factorization and the Rigidity of String Theory,

    N. Arkani-Hamed, C. Cheung, C. Figueiredo, and G. N. Remmen, “Multiparticle Factorization and the Rigidity of String Theory,”Phys. Rev. Lett.132no. 9, (2024) 091601,arXiv:2312.07652 [hep-th]

    arXiv 2024

  42. [42]

    Veneziano variations: how unique are string amplitudes?,

    C. Cheung and G. N. Remmen, “Veneziano variations: how unique are string amplitudes?,”JHEP01(2023) 122,arXiv:2210.12163 [hep-th]

    arXiv 2023

  43. [43]

    On the Inverse Problem in Effective Field Theory,

    F. Calisto, C. Cheung, G. N. Remmen, F. Sciotti, and M. Tarquini, “On the Inverse Problem in Effective Field Theory,”arXiv:2604.15423 [hep-th]

    Pith/arXiv arXiv

  44. [44]

    P. D. B. Collins,An Introduction to Regge Theory and High Energy Physics. Cambridge University Press, 1977

    1977

  45. [45]

    (32) is analogous to a Sommerfeld-Watson transform, again keeping only 7 the leading term in the Legendre polynomial

    Similarly, the inverse transform in Eq. (32) is analogous to a Sommerfeld-Watson transform, again keeping only 7 the leading term in the Legendre polynomial

  46. [46]

    Supersymmetrical Dual String Theory. 2. Vertices and Trees,

    M. B. Green and J. H. Schwarz, “Supersymmetrical Dual String Theory. 2. Vertices and Trees,”Nucl. Phys. B198(1982) 252

    1982

  47. [47]

    Superstring Theory,

    J. H. Schwarz, “Superstring Theory,”Phys. Rept.89 (1982) 223

    1982

  48. [48]

    Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajecto- ries,

    G. Veneziano, “Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajecto- ries,”Nuovo Cim. A57(1968) 190

    1968

  49. [49]

    Reaction amplitude for n-mesons: A generalization of the Veneziano-Bardakçi- Ruegg-Virasoro model,

    Z. Koba and H. B. Nielsen, “Reaction amplitude for n-mesons: A generalization of the Veneziano-Bardakçi- Ruegg-Virasoro model,”Nucl. Phys. B10(1969) 633

    1969

  50. [50]

    Open string amplitudes: singularities, asymp- totics and new representations,

    N. Arkani-Hamed, C. Figueiredo, and G. N. Rem- men, “Open string amplitudes: singularities, asymp- totics and new representations,”JHEP04(2025) 039, arXiv:2412.20639 [hep-th]

    arXiv 2025

  51. [51]

    Symmetries of Bernoulli polynomial se- ries and Arakawa-Kaneko zeta functions,

    P. T. Young, “Symmetries of Bernoulli polynomial se- ries and Arakawa-Kaneko zeta functions,”J. Number Theory143(2014) 142

    2014

  52. [52]

    A connec- tion between power series and Dirichlet series,

    L. M. Navas, F. J. Ruiz, and J. L. Varona, “A connec- tion between power series and Dirichlet series,”J. Math. Anal. App.493(2021) 124541

    2021

  53. [53]

    The Harmonic Logarithms and the Bino- mial Formula,

    S. Roman, “The Harmonic Logarithms and the Bino- mial Formula,”J. Comb. Theory63(1993) 143

    1993

  54. [54]

    Ein Summierungsverfahren für die Rie- mannscheζ-Reihe,

    H. Hasse, “Ein Summierungsverfahren für die Rie- mannscheζ-Reihe,”Math. Z.32(1930) 458. Appendix.—While the construction in this paper is novel, involving the fixed-tMellin transform of the string amplitude—i.e., of the Euler beta integral—and its char- acterization as a deformation of the Riemann zeta func- tion, for the sake of completeness it is worthw...

    1930