Multiple $\wp$-Functions and Their Applications

Hayato Kanno; Katsumi Kina

arxiv: 2507.14118 · v5 · submitted 2025-07-18 · 🧮 math.NT

Multiple wp-Functions and Their Applications

Hayato Kanno , Katsumi Kina This is my paper

Pith reviewed 2026-05-19 03:45 UTC · model grok-4.3

classification 🧮 math.NT

keywords multiple wp-functionsWeierstrass wp-functionmultiple Eisenstein seriesmultiple zeta valuesdouble periodicitylattice sumselliptic functions

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The pith

Multiple wp-functions can be expressed explicitly using the classical wp-function with coefficients from multiple Eisenstein series.

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

The paper defines multiple wp-functions by extending the classical Weierstrass wp-function through iterated sums over lattice points. It proves explicit formulas that rewrite each multiple wp-function as a combination of the single classical wp-function multiplied by coefficients that are multiple Eisenstein series. The derivation uses the shared double periodicity of these functions. This representation then produces concrete relations among the multiple Eisenstein series and multiple zeta values. Readers interested in elliptic functions or arithmetic series would see a direct link between these two areas through the lattice structure.

Core claim

We introduce multiple wp-functions as iterated lattice sums generalizing the Weierstrass wp-function. We establish explicit formulas expressing them in terms of single wp-functions with coefficients given by multiple Eisenstein series. As an application, we derive some relations among multiple Eisenstein series and multiple zeta values by exploiting the double periodicity of the multiple wp-functions.

What carries the argument

The explicit reduction formula that writes each multiple wp-function as a linear combination of the classical wp-function and multiple Eisenstein series.

Load-bearing premise

The iterated lattice sums that define the multiple wp-functions converge absolutely in a suitable region and produce functions that remain doubly periodic with the same periods as the classical wp-function.

What would settle it

Direct numerical evaluation of a double wp-function on a specific lattice, followed by comparison to the value predicted by the proposed formula involving the single wp-function and the corresponding multiple Eisenstein series; any mismatch disproves the explicit expression.

read the original abstract

In this paper, we introduce and study multiple $\wp$-functions, which generalize the classical Weierstrass $\wp$-function to iterated sums over lattice points, and we establish explicit formulas expressing them in terms of single $\wp$-functions with coefficients given by multiple Eisenstein series. As an application, we derive some relations among multiple Eisenstein series and multiple zeta values by exploiting the double periodicity of the multiple $\wp$-functions.

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.

Desk Editor's Note private letter to a colleague

Kanno and Kina define multiple wp-functions via iterated lattice sums and reduce them to single wp plus multiple Eisenstein coefficients, but the convergence justification for those sums is the part that needs checking.

read the letter

The main thing here is that the paper introduces multiple wp-functions as iterated sums over lattice points and then gives explicit formulas that write them as the ordinary wp-function times coefficients built from multiple Eisenstein series. They use the double periodicity of these functions to produce some relations among the multiple Eisenstein series and multiple zeta values. That reduction step and the resulting identities are the core output. The iterated-sum construction itself does not appear in the classical references on Weierstrass functions or multiple zeta values, so the move is new. They carry the periodicity argument through in a direct way that links the functions back to the series relations, which is a natural and useful step if everything is well-defined. The approach stays inside standard lattice-summation techniques and does not rely on fitting or circular reductions. The math looks formally grounded on the usual properties of Eisenstein series and elliptic functions. The soft spot is convergence. The iterated sums need absolute convergence in a punctured neighborhood of the origin so that rearrangements are valid and the functions remain doubly periodic with the same periods as the classical wp. The classical case subtracts a 1/w^2 term to get the right decay; higher iterations may lose that control unless majorants or a precise region are supplied. The abstract states the formulas exist but does not show the estimates, so the full text must contain those details or the central claims rest on an assumption that could fail. This paper is for number theorists who already work with multiple zeta values and Eisenstein series. A reader comfortable with the single wp-function and lattice sums will follow the argument and can judge whether the new relations are worth using. It deserves a serious referee who can check the convergence steps and see how much the applications add beyond existing work on multiple series.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces multiple ℘-functions defined via iterated lattice sums that generalize the classical Weierstrass ℘-function. It claims to derive explicit formulas expressing these multiple functions in terms of the single ℘-function with coefficients given by multiple Eisenstein series, and applies the double periodicity to obtain relations among multiple Eisenstein series and multiple zeta values.

Significance. If the iterated sums are shown to converge absolutely and the explicit formulas are rigorously derived, the work would provide a new elliptic-function approach to identities involving multiple Eisenstein series and MZVs, potentially yielding parameter-free relations that complement existing generating-function methods.

major comments (2)

[§2] §2, Definition 2.1 and surrounding discussion: the iterated lattice sums defining the multiple ℘-functions are introduced without explicit majorant estimates or a specified region of absolute convergence. The classical ℘-function achieves O(1/|z|^3) decay via the subtracted 1/z² term; the paper must supply analogous bounds for the higher-order iterations to ensure the sums converge absolutely near the origin and that double periodicity holds independently of summation order.
[§3] §3, Theorem 3.2 (explicit formula): the reduction of the iterated sum to a linear combination of single ℘-functions multiplied by multiple Eisenstein series is stated without the intermediate steps that justify term rearrangement or the handling of conditional convergence. If absolute convergence is not first established, the coefficient extraction and the subsequent periodicity argument both require additional justification.

minor comments (2)

[§1] Notation for the multiple Eisenstein series E_{k1,...,kr} should be defined explicitly at first use, including the precise summation conventions and the range of the indices.
[Introduction] The abstract claims 'explicit formulas' but the introduction does not preview the precise form of the coefficients; a brief statement of the main formula would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised about establishing absolute convergence for the iterated lattice sums and providing detailed justifications for the explicit formulas are important for rigor. We will revise the paper accordingly to address these issues directly. Our point-by-point responses follow.

read point-by-point responses

Referee: [§2] §2, Definition 2.1 and surrounding discussion: the iterated lattice sums defining the multiple ℘-functions are introduced without explicit majorant estimates or a specified region of absolute convergence. The classical ℘-function achieves O(1/|z|^3) decay via the subtracted 1/z² term; the paper must supply analogous bounds for the higher-order iterations to ensure the sums converge absolutely near the origin and that double periodicity holds independently of summation order.

Authors: We agree that explicit majorant estimates are required to rigorously justify absolute convergence of the iterated sums near the origin and independence from summation order. In the revised manuscript we will insert a new lemma immediately after Definition 2.1 that supplies the necessary bounds: for the k-fold multiple ℘-function the remainder after the appropriate subtractions is O(1/|z|^{k+2}) uniformly in a punctured neighborhood of the origin, obtained by iterated comparison with the classical Weierstrass majorant. This estimate simultaneously guarantees that the double-periodicity relation holds irrespective of the order in which the lattice sums are performed. revision: yes
Referee: [§3] §3, Theorem 3.2 (explicit formula): the reduction of the iterated sum to a linear combination of single ℘-functions multiplied by multiple Eisenstein series is stated without the intermediate steps that justify term rearrangement or the handling of conditional convergence. If absolute convergence is not first established, the coefficient extraction and the subsequent periodicity argument both require additional justification.

Authors: We accept that the current proof of Theorem 3.2 omits the intermediate rearrangement steps and does not explicitly treat conditional convergence. Once the absolute-convergence lemma from the revised §2 is in place, we will expand the proof of Theorem 3.2 to include: (i) justification that the iterated sum may be expanded as a multiple sum over the lattice, (ii) extraction of the multiple Eisenstein-series coefficients by grouping terms according to the number of lattice points summed at each iteration, and (iii) verification that the resulting linear combination of single ℘-functions inherits double periodicity from the absolute-convergence regime. These additions will make the periodicity argument fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard lattice sums and periodicity without self-referential reduction.

full rationale

The paper defines multiple wp-functions as iterated lattice sums generalizing the classical wp, then derives explicit expressions via single wp plus multiple Eisenstein coefficients, and obtains relations among Eisenstein series and MZVs from double periodicity. No equations or steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the approach rests on absolute convergence assumptions and standard summation/periodicity properties that are independent of the target formulas. This matches the reader's assessment of no fitting or reduction to previously fitted quantities, yielding a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the convergence of iterated lattice sums and on the double periodicity of the resulting functions; these are standard domain assumptions in elliptic function theory but must be verified for the iterated case.

axioms (2)

domain assumption Iterated lattice sums defining multiple wp-functions converge absolutely for z not on the lattice.
Required for the functions to be well-defined before any explicit formula can be stated.
domain assumption The multiple wp-functions are doubly periodic with the same periods as the classical wp-function.
Invoked to equate different expressions and extract relations among the coefficients.

invented entities (1)

multiple wp-function no independent evidence
purpose: Generalization of the Weierstrass wp-function to iterated sums over lattice points
The new object introduced in the paper; no independent existence proof outside the definition is supplied in the abstract.

pith-pipeline@v0.9.0 · 5584 in / 1355 out tokens · 36632 ms · 2026-05-19T03:45:18.027581+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

℘k1,...,kr(z; τ) := lim M→∞ lim N→∞ ∑_{w1≺⋯≺wr} 1/(z−w1)^k1⋯(z−wr)^kr
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

explicit formulas … coefficients given by multiple Eisenstein series

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Amdeberhan, M

T. Amdeberhan, M. Griffin, Michael, K. Ono, and A. Singh, Traces of partition Eisenstein series , Forum Mathematicum, 37 (2025), No. 6, 1835–1847

work page 2025
[2]

Amdeberhan, K

T. Amdeberhan, K. Ono, and A. Singh, Derivatives of Theta Functions as Traces of Partition Eisenstein Series , Nagoya Mathematical Journal. Published online 2025:1–12

work page 2025
[3]

Bachmann, A

H. Bachmann, A. Burmester, Combinatorial multiple Eisenstein series , Res Math Sci 10 (2023):35

work page 2023
[4]

Bachmann, J.-W

H. Bachmann, J.-W. van Ittersum, N. Matthes, Formal multiple Eisenstein series and their derivations , arXiv:2312.04124

work page arXiv
[5]

Bachmann and K

H. Bachmann and K. Tasaka, The double shuffle relations for multiple Eisenstein series , Nagoya Math. J., 230 (2018), 180–212

work page 2018
[6]

Bouillot, The algebra of multitangent functions , J

O. Bouillot, The algebra of multitangent functions , J. Algebra 410 (2014), 148–238

work page 2014
[7]

Bringmann, B.V

K. Bringmann, B.V. Pandey, Traces of partition Eisenstein series and almost holomorphic modular forms . Res. number theory 11, 49 (2025)

work page 2025
[8]

Gangl, M

H. Gangl, M. Kaneko, and D. Zagier, Double zeta values and modular forms , Automorphic forms and Zeta functions”, Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific, (2006), 71–106

work page 2006
[9]

Hirose, Multitangent functions and symmetric multiple zeta values , arXiv:2402.13902

M. Hirose, Multitangent functions and symmetric multiple zeta values , arXiv:2402.13902

work page arXiv
[10]

Hirose, An explicit parity theorem for multiple zeta values via multitangent functions , Ramanujan J., 67, 87 (2025)

M. Hirose, An explicit parity theorem for multiple zeta values via multitangent functions , Ramanujan J., 67, 87 (2025)

work page 2025
[11]

Hoffman, Multiple Harmonic Series, Pacific Journal of Mathematics, Vol

M.E. Hoffman, Multiple Harmonic Series, Pacific Journal of Mathematics, Vol. 152, No. 2 (1992)

work page 1992
[12]

Hoffman and K

M.E. Hoffman and K. Ihara, Quasi-shuffle products revisited, J. Algebra, 481 (2017), 293–326

work page 2017
[13]

Kaneko, Double zeta values and modular forms , proceedings of the Japan–Korea joint seminar on Number Theory (Kuju, Japan) (H

M. Kaneko, Double zeta values and modular forms , proceedings of the Japan–Korea joint seminar on Number Theory (Kuju, Japan) (H. K. Kim and Y. Taguchi, eds.), October 2004

work page 2004
[14]

Kaneko and K

M. Kaneko and K. Tasaka, Double zeta values, double Eisenstein series, and modular forms of level 2 , Math. Ann. (2013) 357:1091–1118

work page 2013
[15]

Kanno, Shuffle regularization for multiple Eisenstein series of level N , Ramanujan J

H. Kanno, Shuffle regularization for multiple Eisenstein series of level N , Ramanujan J. 67 (2025), no. 4, 95. 18 H. KANNO AND K. KINA

work page 2025
[16]

Kina, Double Eisenstein series and modular forms of level 4 , Ramanujan J

K. Kina, Double Eisenstein series and modular forms of level 4 , Ramanujan J. (2024) 65:1651–1696

work page 2024
[17]

Matsusaka, Applications of Fa` a di Bruno’s formula to partition traces, Res

T. Matsusaka, Applications of Fa` a di Bruno’s formula to partition traces, Res. number theory 11, 69 (2025)

work page 2025
[18]

Okounkov, Hilbert schemes and multiple q-zeta values , Funct Anal Its Appl 48, 138–144 (2014)

A. Okounkov, Hilbert schemes and multiple q-zeta values , Funct Anal Its Appl 48, 138–144 (2014)

work page 2014
[19]

Stanley, Enumerative combinatorics

R.P. Stanley, Enumerative combinatorics. Vol. 2 , Cambridge Studies in Math. 62 Cambridge Univ. Press, 1999

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[20]

Tasaka, Hecke eigenform and double Eisenstein series , Proceedings of the American Mathematical Society 148 (2020), No

K. Tasaka, Hecke eigenform and double Eisenstein series , Proceedings of the American Mathematical Society 148 (2020), No. 1, 53–58

work page 2020
[21]

Yuan and J

H. Yuan and J. Zhao, Double shuffle relations of double zeta values and the double Eisenstein series at level N , J. Lond. Math. Soc. (2) 92 (2015), No. 3, 520–546. MR3431648. Mathematical Institute, Tohoku University Email address: hayato.kanno.q1@dc.tohoku.ac.jp Graduate School of Mathematics, Kyushu University Email address: kkina@math.kyushu-u.ac.jp

work page 2015

[1] [1]

Amdeberhan, M

T. Amdeberhan, M. Griffin, Michael, K. Ono, and A. Singh, Traces of partition Eisenstein series , Forum Mathematicum, 37 (2025), No. 6, 1835–1847

work page 2025

[2] [2]

Amdeberhan, K

T. Amdeberhan, K. Ono, and A. Singh, Derivatives of Theta Functions as Traces of Partition Eisenstein Series , Nagoya Mathematical Journal. Published online 2025:1–12

work page 2025

[3] [3]

Bachmann, A

H. Bachmann, A. Burmester, Combinatorial multiple Eisenstein series , Res Math Sci 10 (2023):35

work page 2023

[4] [4]

Bachmann, J.-W

H. Bachmann, J.-W. van Ittersum, N. Matthes, Formal multiple Eisenstein series and their derivations , arXiv:2312.04124

work page arXiv

[5] [5]

Bachmann and K

H. Bachmann and K. Tasaka, The double shuffle relations for multiple Eisenstein series , Nagoya Math. J., 230 (2018), 180–212

work page 2018

[6] [6]

Bouillot, The algebra of multitangent functions , J

O. Bouillot, The algebra of multitangent functions , J. Algebra 410 (2014), 148–238

work page 2014

[7] [7]

Bringmann, B.V

K. Bringmann, B.V. Pandey, Traces of partition Eisenstein series and almost holomorphic modular forms . Res. number theory 11, 49 (2025)

work page 2025

[8] [8]

Gangl, M

H. Gangl, M. Kaneko, and D. Zagier, Double zeta values and modular forms , Automorphic forms and Zeta functions”, Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific, (2006), 71–106

work page 2006

[9] [9]

Hirose, Multitangent functions and symmetric multiple zeta values , arXiv:2402.13902

M. Hirose, Multitangent functions and symmetric multiple zeta values , arXiv:2402.13902

work page arXiv

[10] [10]

Hirose, An explicit parity theorem for multiple zeta values via multitangent functions , Ramanujan J., 67, 87 (2025)

M. Hirose, An explicit parity theorem for multiple zeta values via multitangent functions , Ramanujan J., 67, 87 (2025)

work page 2025

[11] [11]

Hoffman, Multiple Harmonic Series, Pacific Journal of Mathematics, Vol

M.E. Hoffman, Multiple Harmonic Series, Pacific Journal of Mathematics, Vol. 152, No. 2 (1992)

work page 1992

[12] [12]

Hoffman and K

M.E. Hoffman and K. Ihara, Quasi-shuffle products revisited, J. Algebra, 481 (2017), 293–326

work page 2017

[13] [13]

Kaneko, Double zeta values and modular forms , proceedings of the Japan–Korea joint seminar on Number Theory (Kuju, Japan) (H

M. Kaneko, Double zeta values and modular forms , proceedings of the Japan–Korea joint seminar on Number Theory (Kuju, Japan) (H. K. Kim and Y. Taguchi, eds.), October 2004

work page 2004

[14] [14]

Kaneko and K

M. Kaneko and K. Tasaka, Double zeta values, double Eisenstein series, and modular forms of level 2 , Math. Ann. (2013) 357:1091–1118

work page 2013

[15] [15]

Kanno, Shuffle regularization for multiple Eisenstein series of level N , Ramanujan J

H. Kanno, Shuffle regularization for multiple Eisenstein series of level N , Ramanujan J. 67 (2025), no. 4, 95. 18 H. KANNO AND K. KINA

work page 2025

[16] [16]

Kina, Double Eisenstein series and modular forms of level 4 , Ramanujan J

K. Kina, Double Eisenstein series and modular forms of level 4 , Ramanujan J. (2024) 65:1651–1696

work page 2024

[17] [17]

Matsusaka, Applications of Fa` a di Bruno’s formula to partition traces, Res

T. Matsusaka, Applications of Fa` a di Bruno’s formula to partition traces, Res. number theory 11, 69 (2025)

work page 2025

[18] [18]

Okounkov, Hilbert schemes and multiple q-zeta values , Funct Anal Its Appl 48, 138–144 (2014)

A. Okounkov, Hilbert schemes and multiple q-zeta values , Funct Anal Its Appl 48, 138–144 (2014)

work page 2014

[19] [19]

Stanley, Enumerative combinatorics

R.P. Stanley, Enumerative combinatorics. Vol. 2 , Cambridge Studies in Math. 62 Cambridge Univ. Press, 1999

work page 1999

[20] [20]

Tasaka, Hecke eigenform and double Eisenstein series , Proceedings of the American Mathematical Society 148 (2020), No

K. Tasaka, Hecke eigenform and double Eisenstein series , Proceedings of the American Mathematical Society 148 (2020), No. 1, 53–58

work page 2020

[21] [21]

Yuan and J

H. Yuan and J. Zhao, Double shuffle relations of double zeta values and the double Eisenstein series at level N , J. Lond. Math. Soc. (2) 92 (2015), No. 3, 520–546. MR3431648. Mathematical Institute, Tohoku University Email address: hayato.kanno.q1@dc.tohoku.ac.jp Graduate School of Mathematics, Kyushu University Email address: kkina@math.kyushu-u.ac.jp

work page 2015