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arxiv: 2604.13365 · v1 · submitted 2026-04-15 · 🧮 math.NT

Representation of Ramanujan's tau function by twisted divisor functions

Tianyu Ni This is my paper

Pith reviewed 2026-05-10 13:07 UTC · model grok-4.3

classification 🧮 math.NT
keywords Ramanujan's tau functiontwisted divisor functionsconvolution sumscusp formsmodular formslevel onearithmetic identities
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The pith

Ramanujan's tau function can be expressed as convolution sums of twisted divisor functions in an infinite family of identities.

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

The paper establishes an infinite collection of explicit identities that write Ramanujan's tau function as sums of products of twisted divisor functions. It obtains these identities by building non-vanishing cusp forms of level one directly from modular forms defined at higher levels. A sympathetic reader would care because tau encodes deep information about partitions and elliptic curves, so new representations could expose previously hidden arithmetic relations. The method supplies concrete formulas rather than existence statements. If the construction works as described, it enlarges the set of known ways to manipulate tau without changing its fundamental definition.

Core claim

We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level 1 cusp forms from modular forms of higher levels.

What carries the argument

The explicit construction of non-vanishing level-1 cusp forms from modular forms of higher levels, which directly produces the identities relating tau to twisted divisor convolutions.

If this is right

  • Tau(n) equals a finite sum of twisted divisor products for each identity in the family.
  • Each such identity supplies an alternative arithmetic expression for the same sequence of tau values.
  • The lifting process from higher-level forms generates new relations among cusp forms at different levels.
  • The family is infinite, so arbitrarily many distinct divisor-sum representations of tau exist.

Where Pith is reading between the lines

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

  • These identities could be used to derive recurrence relations or generating-function identities for tau that were not previously visible.
  • Similar lifting constructions might produce representations for other arithmetic functions attached to cusp forms, such as coefficients of newforms at higher weight.
  • Numerical verification of the identities for large n would provide independent evidence that the level-1 forms constructed in the paper are indeed non-vanishing.

Load-bearing premise

The explicit construction of non-vanishing level-1 cusp forms from modular forms of higher levels is valid and directly produces the claimed infinite family of identities.

What would settle it

A direct computation showing that one of the constructed level-1 cusp forms vanishes identically, or that the corresponding identity for tau fails to hold for some positive integer n.

read the original abstract

We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms of higher levels.

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

0 major / 2 minor

Summary. The paper claims to establish an infinite family of identities expressing Ramanujan's tau function τ(n) via convolution sums of twisted divisor functions. The method proceeds by explicitly constructing non-vanishing level-1 cusp forms from modular forms of higher levels, leveraging the one-dimensionality of the space of weight-12 level-1 cusp forms to recover tau.

Significance. If the explicit construction is valid and produces the claimed identities without circularity or vanishing obstructions, the result would supply new, explicit arithmetic representations for τ(n) that connect divisor sums at twisted levels to the Ramanujan Delta function. This could be useful for both theoretical identities and computational verification in the theory of modular forms.

minor comments (2)
  1. The abstract and method description are high-level; the manuscript would benefit from at least one fully worked example (e.g., the first identity in the family with explicit twisted divisors and numerical verification for small n) to make the construction concrete.
  2. Notation for the twisted divisor functions and the precise twisting characters should be introduced with a dedicated preliminary section or subsection before the main construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee's summary accurately captures the main results concerning the representation of Ramanujan's tau function via convolution sums of twisted divisor functions obtained from explicit constructions of non-vanishing level-1 cusp forms.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's method is an explicit construction of non-vanishing level-1 cusp forms from higher-level modular forms, which is presented as directly producing the claimed infinite family of tau identities via convolution sums of twisted divisor functions. No equations, definitions, or steps are shown that reduce by construction to the inputs (e.g., no self-definitional relations where a parameter is fitted and then renamed as a prediction, no load-bearing self-citations whose uniqueness theorem is invoked to force the result, and no ansatz smuggled via prior work). The one-dimensionality of the weight-12 level-1 cusp form space is a standard external fact compatible with recovering tau, not an internal circularity. The derivation is therefore self-contained against external modular-form benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5308 in / 1025 out tokens · 51356 ms · 2026-05-10T13:07:56.177778+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    The Rankin-Selberg method: a survey

    Daniel Bump. The Rankin-Selberg method: a survey. InNumber theory, trace formulas and discrete groups (Oslo, 1987), pages 49–109. Academic Press, Boston, MA, 1989

    work page 1987

  2. [2]

    Sums involving the values at negative integers ofL-functions of quadratic characters.Math

    Henri Cohen. Sums involving the values at negative integers ofL-functions of quadratic characters.Math. Ann., 217(3):271–285, 1975

    work page 1975

  3. [3]

    Henri Cohen.Number theory. Vol. I. Tools and Diophantine equations, volume 239 ofGrad- uate Texts in Mathematics. Springer, New York, 2007

    work page 2007

  4. [4]

    Springer-Verlag, New York, 2005

    Fred Diamond and Jerry Shurman.A first course in modular forms, volume 228 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2005

    work page 2005

  5. [5]

    El Gradechi

    Amine M. El Gradechi. The Lie theory of certain modular form and arithmetic identities. Ramanujan J., 31(3):397–433, 2013

    work page 2013

  6. [6]

    Cuspidal projections of products of Eisenstein series.J

    William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, and Daozhou Zhu. Cuspidal projections of products of Eisenstein series.J. Number Theory, 234:48–73, 2022

    work page 2022

  7. [7]

    Gross and Don B

    Benedict H. Gross and Don B. Zagier. Heegner points and derivatives ofL-series.Invent. Math., 84(2):225–320, 1986

    work page 1986

  8. [8]

    Subspaces spanned by eigenforms with nonvanishing twisted central l-values.Canadian Journal of Mathematics, page 1–34, 2025.https://doi.org/10.4153/S0008414X25101697

    June Kayath, Connor Lane, Ben Neifeld, Tianyu Ni, and Hui Xue. Subspaces spanned by eigenforms with nonvanishing twisted central l-values.Canadian Journal of Mathematics, page 1–34, 2025.https://doi.org/10.4153/S0008414X25101697. 12 TIANYU NI

    work page doi:10.4153/s0008414x25101697 2025

  9. [9]

    Kohnen and D

    W. Kohnen and D. Zagier. Values ofL-series of modular forms at the center of the critical strip.Invent. Math., 64(2):175–198, 1981

    work page 1981

  10. [10]

    Maass operators and van der Pol-type identities for Ramanujan’s tau function.Acta Arith., 113(2):157–167, 2004

    Dominic Lanphier. Maass operators and van der Pol-type identities for Ramanujan’s tau function.Acta Arith., 113(2):157–167, 2004

    work page 2004

  11. [11]

    Combinatorics of Maass-Shimura operators.J

    Dominic Lanphier. Combinatorics of Maass-Shimura operators.J. Number Theory, 128(8):2467–2487, 2008

    work page 2008

  12. [12]

    216 ofLecture Notes in Mathematics

    Hans Maass.Siegel’s modular forms and Dirichlet series, volume Vol. 216 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great repre- sentative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday

    work page 1971

  13. [13]

    Springer Monographs in Mathematics

    Toshitsune Miyake.Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda

    work page 2006

  14. [14]

    Twisted periods of modular forms.Nagoya Math

    Tianyu Ni and Hui Xue. Twisted periods of modular forms.Nagoya Math. J., 261:e18, 2026

    work page 2026

  15. [15]

    Ramakrishnan and Brundaban Sahu

    B. Ramakrishnan and Brundaban Sahu. Identities for the Ramanujan tau function and certain convolution sum identities for the divisor functions. InHighly composite: papers in number theory, volume 23 ofRamanujan Math. Soc. Lect. Notes Ser., pages 63–75. Ramanujan Math. Soc., Mysore, 2016

    work page 2016

  16. [16]

    R. A. Rankin. The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. (N.S.), 20:103–116, 1956

    work page 1956

  17. [17]

    H. L. Resnikoff. On differential operators and automorphic forms.Trans. Amer. Math. Soc., 124:334–346, 1966

    work page 1966

  18. [18]

    Code for nonsingularity of matrices that are related to twisted periods.https: //github.com/erick-ross/twisted-periods

    Erick Ross. Code for nonsingularity of matrices that are related to twisted periods.https: //github.com/erick-ross/twisted-periods

    work page

  19. [19]

    The special values of the zeta functions associated with cusp forms.Comm

    Goro Shimura. The special values of the zeta functions associated with cusp forms.Comm. Pure Appl. Math., 29(6):783–804, 1976

    work page 1976

  20. [20]

    van der Pol

    Balth. van der Pol. On a non-linear partial differential equation satisfied by the logarithm of the Jacobian theta-functions, with arithmetical applications. I, II.Indag. Math., 13:261–271, 272–284, 1951. Nederl. Akad. Wetensch. Proc. Ser. A54

    work page 1951

  21. [21]

    Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics

    Lawrence C. Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997

    work page 1997

  22. [22]

    Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields

    Don Zagier. Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. InModular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pages 105–169. Lecture Notes in Math., Vol. 627, 1977. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634-0975, USA

    work page 1976