Sums of Hurwitz Class Numbers and newform of weight 2 and level 49

Fangmin Guo

REVIEW 4 major objections 4 minor 18 references

Sums of Hurwitz Class Numbers and newform of weight 2 and level 49

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Pith's one-line read For primes $p\equiv1,2,4\pmod7$, the seven sums $H_{a,7}(p)$ are closed forms in $p$ and the unique $x$ with $p=x^2+7y^2$.

desk verdict Main formula likely right, but Lemma 4.2 is false as printed and the proof relies on unaudited Sturm checks. read the letter →

arxiv 2411.16894 v1 pith:TRCI4MUP submitted 2024-11-25 math.NT

Fangmin Guo This is my paper · ORCID

classification math.NT MSC 11F3711F1111E4111F30

keywords Hurwitzclassnumbersmixedmockmodularformsnewform49.2.a.arepresentationp=x^2+7y^2Sturmboundweight-2FouriercoefficientsCM

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 aims to establish exact prime-index formulas for the seven sums of Hurwitz class numbers $H_{a,7}(p)=\sum_{t\equiv a\pmod7}H(4p-t^2)$. The main theorem states that for every prime $p>2$ with $p\equiv1,2,4\pmod7$, these sums are completely determined by $p$ and by the unique positive integer $x$ in $p=x^2+7y^2$: each equals a fixed rational function of $p$ plus a term $\tfrac12\chi_{-7}(x)x$ or $\tfrac14\chi_{-7}(x)x$. A reader should care because it closes a table of cases left open by earlier work, and because it shows the prime values of these class-number sums directly encode a binary quadratic form representation. If the formulas are right, no unknown quantity remains in the correction term for these primes.

What carries the argument

The mechanism is the completion of mixed mock modular forms. The generating function $H_{m,7}(\tau)=\sum_n H_{m,7}(n)q^n$ is not modular, but Theorem 2.7, a Rankin-Cohen bracket completion identity, upgrades it to a quasimodular form once the computable series $\frac12\Lambda_{1,m,7}|U_4$ is added. Acting by sieving operators or by the character twist $\chi_{-7}^2$ puts the result in the finite-dimensional space $M_2(\Gamma_0(4\cdot49)\cap\Gamma_1(7))$ or $M_2(\Gamma_0(4\cdot49))$, where Theorem 3.5 writes it as a combination of the divisor-sum series $D$ and the newform $G$ (label 49.2.a.a, the weight-2 CM newform of level 49). The bridge to binary quadratic forms is Lemma 4.2, $\Psi_7(\chi_{-7},\tau)=G-G|U(2)+4G|U(4)$, where $\Psi_7$ is the $\theta$ series $\frac12\sum_n\bigl(\sum_{x^2+7y^2=n}\chi_{-7}(x)x\bigr)q^n$; here $U(M)$ keeps only Fourier indices divisible by $M$, so for odd primes $p$ the coefficient of $q^p$ on the right is exactly $\chi_{-7}(x)x$. Lemma 4.3 then supplies the uniqueness of $x$.

What would settle it

Run an independent Sturm-bound check of Lemma 4.2 and Theorem 3.5 (the first 56 coefficients and the first 57 or 337 coefficients, respectively). A faster spot check is $p=29$: the theorem predicts $H_{0,7}(29)=8$, $H_{\pm1,7}(29)=10$, $H_{\pm2,7}(29)=8$, and $H_{\pm3,7}(29)=7$, each of which can be evaluated directly from the definition $H_{a,7}(29)=\sum_{t\equiv a\pmod7}H(116-t^2)$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.4: for every prime $p>2$ with $p\equiv1,2,4\pmod7$ there is a unique positive integer $x$ with $p=x^2+7y^2$, and the seven sums $H_{a,7}(p)$ are explicit rational functions of $p$ and $x$. Writing $\chi=\chi_{-7}$ for the non-principal real character modulo 7, the values are $$H_{0,7}(p)=\frac{p+1}{4}+\frac12\chi(x)x,$$ $$H_{\pm1,7}(p)=\begin{cases}\frac{p+1}{3},&p\equiv1\\ \frac{7p+7}{24}+\frac14\chi(x)x,&p\equiv2\\ \frac{p+1}{4}-\frac12\chi(x)x,&p\equiv4\end{cases}\pmod7,$$ $$H_{\pm2,7}(p)=\begin{cases}\frac{7p-17}{24}+\frac14\chi(x)x,&p\equiv1\\ \frac{p+1}{4}-\frac12\chi(x)x,&p\equiv2\\ \frac{p-2}{3},&p\equiv4\end{cases}\pmod7,$$ $$H_{\pm3,7}(p)=\begin{cases}\frac{p+1}{4}-\frac12\chi(x)x,&p\equiv1\\ \frac{p-2}{3},&p\equiv2\\ \frac{7p+7}{24}+\frac14\chi(x)x,&p\equiv4\end{cases}\pmod7.$$ The closing table supplies the values for $p\equiv3,5,6\pmod7$, where the correction term is absent. The discovery is that one correcting expression $\chi_{-7}(x)x$ accounts for all residue classes that admit the representation, with coefficients $\pm\frac12$ or $\pm\frac14$ depending on $a$ and on $p\bmod7$.

Load-bearing premise

The entire derivation hinges on the claim that two specific modular forms agree because their first 56 Fourier coefficients match; if that coefficient computation, or the longer Sturm-bound checks behind Theorem 3.5, contains even one arithmetic error, the prime-term formulas collapse.

Editorial extensions

If this is right

For every prime $p\equiv1,2,4\pmod7$, finding the unique $x$ in $p=x^2+7y^2$ immediately yields all seven values $H_{a,7}(p)$; no other input is needed.
For $p\equiv3,5,6\pmod7$, the correction term vanishes and each $H_{a,7}(p)$ is a fixed rational function of $p$ alone, exactly as tabulated at the end of the paper.
The $p$-th coefficient of newform 49.2.a.a is $\frac12\sum_{x^2+7y^2=p}\chi_{-7}(x)x$, so the newform's prime arithmetic is governed by this representation.
The formulas complete the $m=7$ cases left open in earlier studies and confirm that the $\chi(x)x$ pattern observed for $M=6,8$ persists for $M=7$.

Reading between the lines

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

Editorial extension: the same finite Sturm-bound proof strategy should yield explicit formulas for $H_{m,M}(p)$ for other fixed $M$, provided the relevant CM newform and its $U$-operator combination are identified.
Editorial extension: the appearance of $G-G|U(2)+4G|U(4)$ rather than $\Psi_7$ itself suggests that for general level the natural error term is a $U(2)$/$U(4)$-modified CM newform; testing this for other discriminants would test the pattern.
Editorial extension: since the correction term is either $\pm\frac12\chi_{-7}(x)x$ or $\pm\frac14\chi_{-7}(x)x$, the sums $H_{a,7}(p)$ as $p$ varies inherit the sign oscillations of $\chi_{-7}(x)$; studying their average over primes is a natural question the paper does not take up.

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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

Here's my read. The main formula for H_{a,7}(p) is very likely correct, but the paper as printed has a real flaw in its bridge lemma. Under the authors' own definition of the U-operator (Definition 2.4(1), keeping a(Mn)), the identity in Lemma 4.2, Psi_7(chi_-7,tau) = G - G|U(2) + 4G|U(4), fails already at the q^1 coefficient. The left side has coefficient 1, the right side has -4. The stress-test note is right. The likely fix is to use the dilation operators V_2 and V_4 instead; since both U_d and V_d kill q^p for odd primes p, the prime-coefficient extraction in Section 4 survives. So the theorem may be true, but the proof as written is internally inconsistent.

What the paper does well: it organizes the seven residue classes cleanly, gives a useful table for all p mod 7, and makes the connection to the CM newform 49.2.a.a explicit. The overall method—completing the mixed mock modular form, sieving into a finite-dimensional space, then applying Sturm bounds—is standard and appropriate. The exposition is readable, if a little rough.

The honest concerns beyond Lemma 4.2: (1) All Sturm-bound checks are asserted, not shown. No code, no tables, no way to audit the 56- and 337-coefficient computations. For a paper whose proof rests on those checks, that is a significant reproducibility gap. (2) The statement of Theorem 4.4 is truncated in the middle of the formulas—the case 'a = ±3, p ≡ 4 mod ...' is cut off, and the table has similar truncations. The main theorem is not fully displayed. (3) Novelty is genuinely unclear. The introduction says [4] and [5] already proved the m=7 conjectures; what is new here is the chi_-7(x)x phrasing and the newform identification, which is a nice translation but not a new result unless the earlier proofs omitted these exact expressions. The paper does not spell out the difference.

Overall, this paper deserves a serious referee because the main result is plausible and the flaw is likely fixable. But I would not accept it as is. The referee should ask for a corrected Lemma 4.2 (or a justification that the U-definition was misprinted), for the actual verification data, and for a clear statement of what was not already in the literature. If those come back, the paper could be a solid contribution to the H_{a,M} table.

Recommendation: send it to review, with a note to the referee to check Lemma 4.2 carefully. I would not cite it in its current form.

Referee Report

4 major / 4 minor

Summary. The paper claims to determine all sums of Hurwitz class numbers H_{a,7}(p) for primes p>2 with p≡1,2,4 mod 7, expressing them in terms of p and the unique positive integer x with p=x^2+7y^2. The method follows Zindulka's framework: the generating series H_{m,7} is completed as a mixed mock modular form, after sieving and twisting it becomes a weight-2 modular form on a congruence subgroup of level 196, and the resulting identities are proved by Sturm-bound coefficient checks. The error term is identified with the Fourier coefficients of the CM newform 49.2.a.a, connected to the earlier form Ψ_7(χ_{-7},τ) by a linear identity (Lemma 4.2).

Significance. If correct, the paper fills the remaining open cases of the M=7 Hurwitz-class-number sum problem and exhibits the same pattern found by Zindulka for M=6,8, namely an error term χ(x)x coming from a CM form attached to the quadratic form x^2+ny^2. A notable strength is that no free parameters are fitted: the right-hand sides are fixed modular-form identities. The link between the class-number sums and the newform 49.2.a.a is a natural and potentially useful observation. However, the verification currently rests on unshown coefficient checks, and one of the key asserted checks is demonstrably wrong as printed, so the paper needs substantial correction before the claims can be accepted.

major comments (4)

[Lemma 4.2] Lemma 4.2 is false under the paper's own Definition 2.4(1). With G=q+q^2-q^4-3q^8-3q^9+..., the q^1 coefficient of G-G|U_2+4G|U_4 is 1-1+4(-1)=-4, whereas Ψ_7(χ_{-7},τ) has q^1 coefficient 1 (the only solutions of x^2+7y^2=1 are x=±1,y=0 and the contribution is (1/2)(χ(1)·1+χ(-1)·(-1))=1). Thus the asserted 56-coefficient check cannot have passed. The identity appears to become correct when the standard dilation operators V_2,V_4 are used, and since both U_2/V_2 and U_4/V_4 annihilate q^p for odd primes p, the final prime-coefficient formulas may survive the correction. Still, the bridge lemma as printed is invalid and must be fixed and re-verified.
[Theorem 4.4] The main theorem is internally inconsistent and incomplete. The case a=±2, p≡1 mod 7 appears in both the second and the fourth displayed cases, with different values: (p+1)/4 - (1/2)χ_{-7}(x)x in the former and (7p-17)/24 + (1/4)χ_{-7}(x)x in the latter. The table at the end of Section 4 gives only the latter for (m,p)=(±2,1). Several congruence conditions are truncated ("p≡4 mod" instead of "p≡4 mod 7"), and no formula is given for a=±1, p≡1 mod 7, which the table lists as (p+1)/3. The main theorem must be restated consistently with the table and with the identities proved in Theorem 3.5.
[Theorem 3.5 and Lemma 4.2 proofs] The proof of Theorem 3.5 and the proof of Lemma 4.2 consist solely of assertions that the first 57 or 337 coefficients coincide, with no tables, code, or reproducible computation. Given that the analogous assertion in Lemma 4.2 is actually false as written, this style of verification is not acceptable in its current form. The author should provide the computed coefficients, a transcript of the calculation, or explicitly checkable data, and should re-run the checks after correcting the operators in Lemma 4.2.
[Theorem 3.5, third block] The displayed equation for (Hθ_{3,7})|U_4|S_{7,4} has right-hand side 7/24 D|S_{7,4} + 1/8 G|S_{7,1}, but the left-hand side is sieved by S_{7,4}. This is almost certainly a typo for 1/8 G|S_{7,4}; as printed, the identity is malformed and should be corrected in the revision.

minor comments (4)

[Throughout] The text contains numerous typographical errors, including "W e", "consi der", "with with", and missing congruences such as "mod 7" in several places; a careful proofreading pass is needed.
[Definition 2.4(2)] The V_M operator is defined as (f|V_M)(τ)=∑ a(Mn)q^{Mn}, which coincides with the sieving operator S_{M,0} rather than with the standard dilation V_M f(q)=f(q^M). This ambiguity likely contributed to the error in Lemma 4.2; the definition should be clarified and made consistent with Proposition 2.5 and with the later usage.
[Proposition 2.5 proof] In the proof of Proposition 2.5(1) the line "Therefore M U_M = Γ_1αΓ_2" is imprecise; the displayed double-coset expression should be normalized consistently so that the operator relation U_M = ... or M^{-1}Γ_1αΓ_2 is stated unambiguously.
[Final table] The table at the end of Section 4 would be easier to read with parentheses around the rational expressions (e.g., (p+1)/4) and with χ consistently written as χ_{-7}(x); as printed, some entries are ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation relies on independent finite q-expansion checks and external CM/modular-form facts, not on fitting the claimed prime formulas.

full rationale

The central claim, Theorem 4.4, is obtained by first proving modular-form identities (Theorem 3.5) whose left sides are defined directly from the Hurwitz class number sums H_{m,7}(n) and whose right sides are explicit combinations of divisor sums and the newform G = 49.2.a.a; the identities are verified by comparing coefficients up to the Sturm bound (337 coefficients for Γ_0(4·49)∩Γ_1(7), 57 for Γ_0(4·49)). No parameter is fitted to the target prime formulas: the coefficient of q^p on the right-hand side is computed from the independently defined series D, G, and Ψ_7(χ_{-7},τ), and the integer x is defined by the independent representation p = x^2 + 7y^2 (Lemma 4.3, cited to Cox). Lemma 4.2, which expresses Ψ_7 as G − G|U(2) + 4G|U(4), is used as a bridge and is asserted by a Sturm-bound check rather than derived symbolically; this is an omitted-computation or correctness risk, and a skeptical check against the paper's own Definition 2.4(1) suggests the q-coefficient identity may not hold with the U-operators as defined. But even if that check is wrong, the defect is a false or unverified intermediate assertion, not circularity: the bridge identity is not assumed as the target formula and is not equivalent to Theorem 4.4 by construction. The paper also relies on external facts (the newform data of 49.2.a.a, the Sturm bound, and Cox's unique-representation lemma); these are independent inputs, not self-citations from the author. No self-definitional, fitted-input, self-citation, uniqueness-imported, or renaming circularity is present.

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

No new free parameters or invented entities appear; the constants in the formulas are determined by the modular form decomposition, not fitted. The argument imports standard machinery (mock modular forms, Sturm bound, Mertens' Lambda decomposition, CM theory of 49.2.a.a). The main unverified burden is the finite coefficient checks in Theorem 3.5 and Lemma 4.2, which are computational rather than axiomatic.

assumptions (5)

standard math Zagier's theorem: H(q) is a weight 3/2 mock modular form with shadow theta (used in Section 2).
Foundation for treating H_{m,M} as mixed mock modular forms; the paper cites [11] and does not reprove it.
standard math Kane-Pujahari Theorem 2.7: the completed expression ([H, theta_{m,M}]_k + 2^{-1-2k} binom(2k,k) Lambda_{2k+1,m,M})|U4 is modular or quasimodular.
Core structural tool that makes the subsequent sieving argument possible; cited from [14].
standard math Mertens' Proposition 3.1 formula for Lambda_{2k+1,m,p}|U(4) in terms of D^{(p,a)}_l series.
Load-bearing for the computable part of the coefficient extraction; cited from [16].
standard math Sturm bound theorem (Proposition 3.3) and the index computation giving bounds 336 and 56.
Justifies proving identities by comparing finitely many Fourier coefficients; the actual coefficient comparisons are not shown.
domain assumption CM property of newform 49.2.a.a: its prime coefficients connect to x in p = x^2 + 7y^2 via Lemma 4.2 and Cox [2].
The chi_{-7}(x)x formula is imported from the known CM theory of this newform, not derived from first principles here.

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Cite this review

Pith. "Pith review of Sums of Hurwitz Class Numbers and newform of weight 2 and level 49." pith.science (2026). https://pith.science/paper/TRCI4MUP

@misc{pith2026241116894,
  author       = {Pith},
  title        = {Pith review of: Sums of Hurwitz Class Numbers and newform of weight 2 and level 49},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRCI4MUP}},
  note         = {Machine review of arXiv:2411.16894}
}

abstract

We consider sums of Hurwitz class number $H_{m,M}(n)=\sum_{t\equiv m (\text{mod} M)}{H(4n-t^2)}$, where $H(N)$ denotes the Hurwitz class number. In this article, we consider the case of $M=7$. By completing the mixed mock modular form generated by $H_{m,7}(n)$, We obtain the formula of modular forms consist of a computable part and a part from newform 49.2.a.a whose prime terms of Fourier expansion has a connection with with $p=x^2+7y^2$ $(p\equiv 1,2,4 \mod 7)$.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Sums of class numbers and mixed mock modular forms

KATHRIN BRINGMANN and BEN KANE. Sums of class numbers and mixed mock modular forms. Mathematical Proceedings of the Cambridge Philosophical S ociety, 167(02):321–333, June 2018

work page 2018

[2] [5]

Michael H. Mertens. Eichler–selberg type identities fo r mixed mock modular forms. Advances in Mathe- matics, 301:359–382, October 2016

work page 2016

[3] [1]

History of the theory of numbers

Leonard Eugene Dickson. History of the theory of numbers . Dover Publications, 2005

work page 2005

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Cox; with contributions by Roger Lipsett

David A. Cox; with contributions by Roger Lipsett. Primes of the form x ² + ny ². AMS Chelsea Publish- ing/American Mathematical Society, 2022

work page 2022

[5] [3]

Calkin, Timothy B

Brittany Brown, Neil J. Calkin, Timothy B. Flowers, Kevi n James, Ethan Smith, and Amy Marguerite Irwin Stout. Elliptic curves, modular forms, and sums of hurwitz c lass numbers. Journal of Number Theory , 128:1847–1863, 2008

work page 2008

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Sums of hurwitz class numbers, cm mo dular forms, and primes of the form x2 + ny2, 2024

Mikul´ aˇ s Zindulka. Sums of hurwitz class numbers, cm mo dular forms, and primes of the form x2 + ny2, 2024

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The L-functions and modular fo rms database, home page of newform orbit 49.2.a.a, 2024

The LMFDB Collaboration. The L-functions and modular fo rms database, home page of newform orbit 49.2.a.a, 2024. [Online; accessed 26 November 2024]

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Quantum black holes, wall crossing, and mock modular forms, 2014

Atish Dabholkar, Sameer Murthy, and Don Zagier. Quantum black holes, wall crossing, and mock modular forms, 2014

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