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REVIEW 3 major objections 4 minor 42 references

The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line

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Pith's one-line read The paper proves that the Epstein zeta-function of every positive definite binary quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line Re s = 1/2.

desk verdict First positive-proportion zero result for Epstein zeta-functions, but the proof leans on an unproved general-case additive lemma and needs a serious referee. read the letter →

arxiv 2411.18492 v1 pith:AM72IHUF submitted 2024-11-27 math.NT

I.S. Rezvyakova This is my paper · ORCID
classification math.NT MSC 11M4111M26
keywords Epsteinzeta-functionHeckeL-functionscriticallinezerospositiveproportionoflinearcombinationshiftedconvolutionsumsRiemannmollifiermethod
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 establishes that the Epstein zeta-function associated with a positive definite binary quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line. More generally, it proves that any real-coefficient linear combination of distinct Hecke L-functions of an imaginary quadratic field has $N_0(T) \gg T \log T$ zeros on the segment $\{s = 1/2 + it,\, T \le t \le 2T\}$. The point of the claim is that such linear combinations have functional equations but lack Euler products, so they can have zeros away from the critical line; the result says a positive fraction of their zeros nonetheless lie exactly on it. This extends earlier positive-proportion results from combinations made only of complex Hecke characters to combinations that also contain real Hecke characters.

What carries the argument

The central object is the mollified sign-change integral: on a set of $t$ where one Hecke L-function dominates the others, the proof compares $I(t,H)=\int_{\mathcal{H}_t} F(t+u)\,|\eta_j(1/2+i(t+u))|^2\, e^{(\pi/2-1/T)(t+u)}\,du$ with $J(t,H)=\int_{\mathcal{H}_t} |F(t+u)\eta_j^2(1/2+i(t+u))|\,e^{(\pi/2-1/T)(t+u)}\,du$; whenever $J(t,H)>|I(t,H)|$, the function changes sign and an odd-order zero occurs. The new machinery enters in bounding the non-diagonal term of the second moment, which is governed by the shifted convolution sum $S=\sum_{m_2n_2-m_1n_1=l} r(n_2)r(n_1)$ for coefficients $r(n)$ of a real-character Hecke L-function. Lemma 13 gives the main term $\sigma(l,m_1,m_2)\pi^2 h^2(-D) N/m_2$ with error $N^{11/13+\varepsilon} m_1^{9/13} m_2^{-2/13}$, and the proof splits the associated Dirichlet series into four classes of summation parameters, shows the two cross-class terms cancel, and estimates the remaining diagonal classes through a series of multiplicative-function bounds.

What would settle it

Compute the shifted sum $S=\sum_{m_2n_2-m_1n_1=l,\, n_1\le N} r(n_2)r(n_1)$ for a small example such as $D=15$ with real characters $\chi_3$ and $\chi_5$, taking $m_1=2$, $m_2=3$, $l=1$, and $N=10^5$, and compare it with the claimed main term $\sigma(1,2,3)\pi^2 h^2(-15) N/3$; any deviation beyond the stated $N^{11/13+\varepsilon}2^{9/13}3^{-2/13}$ would refute Lemma 13 and hence the proof of Theorem 1.

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Extended reading notes

Core claim

Theorem 1 states that for a linear combination $F(s)=\sum_{j=1}^P c_j L_j(s)$ formed with real coefficients $c_j$ from $P$ distinct Hecke L-functions attached to ideal class group characters of $\mathbb{Q}(\sqrt{-D})$, the number $N_0(T)$ of zeros on $\{s=1/2+it,\, T\le t\le 2T\}$ satisfies $N_0(T)\gg T\log T$. The Epstein zeta-function of a positive definite binary quadratic form with integer coefficients is such a linear combination, so it inherits the same positive-proportion property. The novelty is that the combination may contain L-functions of real Hecke characters, which split as products of two Dirichlet L-functions; for these the shifted convolution problem acquires a nonzero main term, and the proof must control that term rather than discard it as small.

Load-bearing premise

The load-bearing premise is that the shifted convolution sum $S$ in Lemma 13 has the stated asymptotic with the stated error for every coprime pair $(m_1,m_2)$; the author does not carry out the general-case proof in this preprint, so the formula is taken on trust.

Editorial extensions

If this is right

  • Every Epstein zeta-function of an integral positive definite binary quadratic form has $N_0(T)\gg T\log T$ zeros on the critical line, matching the order of its total zero count.
  • The positive-proportion theorem holds for the broad class of real-coefficient linear combinations of distinct Hecke L-functions, including real Hecke characters, not only for the Epstein cases.
  • The result is unconditional, in contrast to the conditional almost-all-on-the-line statement for Epstein zeta-functions that relies on a generalized Riemann hypothesis and a pair-correlation conjecture.
  • The same framework gives a corresponding density bound for zeros of each constituent Hecke L-function near the critical line, so almost all non-trivial zeros of those functions lie within $O(\varphi(T)/\log T)$ of $\operatorname{Re} s=1/2$ for any $\varphi(T)\to\infty$.
  • Combined with the functional equation, the proof confirms that a positive proportion of zeros of these non-Euler-product combinations remain on the critical line even though a comparable number may lie off it.

Reading between the lines

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

  • Editorial inference: the structural cancellation between the two cross-class terms in the non-diagonal contribution is likely the reason real Hecke characters do not destroy the positive-proportion phenomenon; a similar cancellation could be sought in other linear combinations of L-functions whose constituents share a partial Euler product.
  • Editorial inference: the method suggests a testable route to quantitative proportions: extracting the constants from the estimates (8)-(10) and the auxiliary lemmas would give an explicit lower bound in $N_0(T)\gg T\log T$, which the paper does not state.
  • Editorial inference: a numerical evaluation of the shifted convolution sum for small coprime pairs $(m_1,m_2)$ could serve as a targeted check of Lemma 13, since the general-case proof is not carried out in the preprint.
  • Editorial inference: the techniques likely extend to real-coefficient combinations of higher-degree L-functions whose Euler-product factors agree on a positive proportion of primes, provided an analogous shifted-convolution main term can be isolated.
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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

3 major / 4 minor

Summary. The paper proves Theorem 1: if F(s) is a real-coefficient linear combination of P distinct Hecke L-functions attached to ideal class group characters of Q(√−D), then a positive proportion of its non-trivial zeros lie on the critical line, quantitatively N0(T) ≫ T log T. Since the Epstein zeta-function of a binary positive definite integral quadratic form is such a linear combination, the stated result for Epstein zeta-functions follows. The proof follows Selberg's method: a mollifier approximating L^{-1/2} is used, the interval (T,2T) is partitioned according to which summand dominates log |L_j(1/2+it)|, and the problem is reduced to the mean estimates (8)–(10). Those estimates are obtained from a diagonal term and a non-diagonal term; the latter is controlled by a shifted convolution sum with coefficients r(n) of L_{χ_{d1}}L_{χ_{d2}}. The main unresolved point in the manuscript is that the key shifted-convolution lemma, Lemma 13, is proved only for m1=m2=1, while the proof in §8 uses the general coprime case without supplying the required derivation.

Significance. If the proof is completed, this is a substantial result: it would give the first positive-proportion-on-the-critical-line statement for the Epstein zeta-function of binary forms and, more generally, for linear combinations of Hecke L-functions that include real Hecke characters, extending the author's earlier work for complex characters. The paper correctly identifies the new difficulty: for real Hecke characters the non-diagonal term is of the same order as the diagonal term, and the main term in the additive problem must be controlled by the arithmetic factor σ(l,m1,m2). The overall strategy is credible and the lengthy §8 estimation, if supplied with a valid Lemma 13, is plausible. However, the manuscript as written is not self-contained at exactly the load-bearing step, so the central claim is not yet fully supported.

major comments (3)
  1. [§7, Lemma 13] The shifted-convolution estimate is load-bearing for the whole non-diagonal part, but its proof is not contained in the manuscript. The text states that the result in [30] is correct only when m1=m2=1 and that the general case 'requires operating with the precise value of the parameter σ from [17, p.278]'; no such general derivation is carried out. This matters because in §8 one has m1=ν1ν4/q and m2=ν2ν3/q with (m1,m2)=1, and these integers are not generally 1. Lemma 13 is then used in Corollary 1 to meromorphically continue D_{m1,m2}(s,l) and to extract its residue at s=1; the estimate of J2(1,θ) and the key bounds (8)–(10) depend on it. Unless the promised general proof is supplied, Theorem 1 is not established.
  2. [§3, estimates (11)–(13)] The value-distribution estimates (11)–(13) for log |L_j(1/2+it)| are imported from [28] with only a 'skeleton' reference, but [28] treats complex ideal class group characters, whereas Theorem 1 requires pairs involving real Hecke characters, for which L(s) is a product of two Dirichlet L-functions. These estimates are used to define the sets S_j on which one summand dominates and to conclude that the sign changes found by comparing I(t,H) and J(t,H) actually produce zeros on the critical line. The paper should either prove the real-character case or quote a precise published statement that covers it.
  3. [§4, Lemmas 3 and Theorem 2] Lemma 3, which converts the mean-square estimates on F(t) and I(t,H) into integrals of G(y), is stated with a proof 'contained in [25]' and is not reproduced. Theorem 2 is described as 'the core of the work' and its proof in §§6 and 8 relies on the unproved Lemma 13 and on several auxiliary statements whose derivations are sketched rather than fully written. This is acceptable for published references in a specialized journal, but the dependency chain should be made explicit so that the reader can verify that no statement from [25], [28], or [30] is being used in a regime outside its hypotheses.
minor comments (4)
  1. [§2] There are numerous typos: 'principle ideal' should be 'principal ideal', 'posess' should be 'possess', and 'the filed' should be 'the field'.
  2. [§5, Lemma 14 statement] In the statement of Lemma 14 the left-hand side uses the variable s while the four summands on the right are written in terms of w; the same letter should be used consistently.
  3. [§3, notation of φ(u)] The function φ(u) is defined at the end of §3 as 1+cos4δ/(cos4δ+u^{-4}), but the notation 'cos4δ' is ambiguous; it would be clearer to write cos(4δ) or (cosδ)^4, depending on intent.
  4. [General] The typography of the references is inconsistent (for example, reference [3] has a repeated author line and [29] has an incomplete English title). A careful copyedit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation does not assume its conclusion; auxiliary appeals to the author's earlier papers are parameter-free published lemmas, and the unproved general case of Lemma 13 is a completeness gap, not a circular reduction.

full rationale

The paper's central claim (Theorem 1) is proved by Selberg's mollifier method: zeros on the critical line are detected through sign changes of a real-valued function, and the proof reduces to establishing the estimates (8)-(10). These estimates are derived from Lemma 3 and Theorem 2, where Lemma 3 is quoted from the author's earlier paper [25] and the complex-character cases are taken from [28]. None of these quoted results assumes Theorem 1 or encodes the target proportion of zeros; they are independent published statements with proofs elsewhere, and their hypotheses do not include the conclusion being proved. The one load-bearing citation that deserves scrutiny is Lemma 13, whose proof is said to be given in [30] only for m1=m2=1, with the general case to be obtained by modifying that proof using a parameter from [17, p.278]; the preprint does not carry out that modification. That is an incompleteness or correctness risk in the non-diagonal estimate of the additive problem, not a circularity: the formula (52) for sigma(l,m1,m2) is not equivalent to the conclusion N0(T) >> T log T, and the paper never assumes the theorem it is proving. No self-definitional step, fitted parameter disguised as a prediction, or uniqueness theorem imported from the authors' prior work appears in the derivation. Under the required standard of exhibiting a specific reduction of the conclusion to the inputs, no circular step can be identified.

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

The paper introduces no new objects beyond standard L-functions, Hecke characters, and Dirichlet polynomials. The proof parameters X and H are chosen inside admissible ranges (X <= T^{1/50}, 1/log T <= H <= (log log T/log T)^{2/3}) and are not fitted to data or to the target claim. The main axiomatic load is the importation of several technical estimates from the author's earlier published work, plus standard background facts on Hecke L-functions.

assumptions (5)
  • standard math Functional equation and modularity for Hecke L-functions (Lemma 1)
    Invoked in Section 2 and used throughout; standard results due to Hecke, as compiled in [11], [13], [15].
  • standard math Kronecker factorization of real Hecke L-functions into two Dirichlet L-functions
    Used in Section 2; standard quadratic field theory.
  • standard math Zero-free regions for the Dirichlet series D(s) and related L-functions used in contour shifts
    Used in Lemma 11 and Lemma 12 to deform contours to the left; standard consequences of known zero-free regions, not proved in this paper.
  • domain assumption Value-distribution estimates (11)-(13) for real Hecke characters
    Quoted from [28] with only a skeleton of proof referenced; these estimates are load-bearing for the partition of (T,2T) into sets S_j in Section 3.
  • domain assumption Shifted convolution asymptotic of Lemma 13 for general m1,m2
    Stated with proof said to follow from [30] only in the case m1=m2=1, with a correction via [17] not carried out in the text; feeds Corollary 1 and the non-diagonal estimate in Section 8.

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Pith. "Pith review of The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line." pith.science (2026). https://pith.science/paper/AM72IHUF

@misc{pith2026241118492,
  author       = {Pith},
  title        = {Pith review of: The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AM72IHUF}},
  note         = {Machine review of arXiv:2411.18492}
}
read the original abstract

It is proved that the Epstein zeta-function corresponding to a binary positive definite quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line.

Discussion (0). Continue with ORCID to comment.

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