{"id":"f6fd03e3-1467-48ee-83d4-b1239a324fe4","arxiv_id":"2411.18492","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any real linear combination of distinct Hecke L-functions over an imaginary quadratic field, including Epstein zeta-functions of binary quadratic forms, a positive proportion of non-trivial zeros lies on Re s = 1/2.","lead":"A number theory proof claims that Epstein zeta-functions attached to binary quadratic forms, and more general combinations of Hecke L-functions, have a positive share of their zeros exactly on the critical line Re s = 1/2. The result extends Selberg's classical method to these degree-two objects and strengthens evidence that non-Euler-product zeta-like functions can still have most of their zeros on the critical line.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-diagonal estimate rests on Lemma 13, whose proof is supplied only for m1=m2=1 and deferred to [30]; the general coprime m1,m2 case is left unproved, so Theorem 1 is not fully supported as written.","rationale":"The reader identified Lemma 13 as the weakest assumption, and I agree. The entire non-diagonal analysis in §8 is channelled through the analytic continuation of D_{m1,m2}(s,l) obtained in Corollary 1, which relies directly on the asymptotic formula of Lemma 13 for general m1,m2. The manuscript states explicitly that the proof in [30] covers only m1=m2=1 and that the general case requires a modification using a parameter from [17, p.278], but the modification is not carried out. This is not a mathematical contradiction; it is a missing verification of a central technical input. The rest of the paper contains many detailed estimates, including lemmas 4 through 12 and the treatment of the diagonal and non-diagonal terms, and the overall strategy is coherent. However, because the unproved lemma is load-bearing, the correct verdict remains conditional rather than accept. My recommendation is therefore unchanged from the reader's CONDITIONAL verdict: the author should provide the complete proof of Lemma 13, or give precise corrected references with the general case worked out, before the theorem can be regarded as fully established.","tokens_in":58815,"tokens_out":3736,"duration_ms":37233,"concrete_test":"Independently re-derive Lemma 13 for general coprime m1,m2 following [30] with the parameter σ from [17, p.278], and verify that the main term is exactly (52), especially by computing the complete q-sum in the four classes Q_{1,1}, Q_{2,2}, Q_{1,2}, Q_{2,1} of Lemma 14. If the derivation introduces extra factors such as χ_{d1}(m1)χ_{d2}(m2) or an altered local density in σ(l,m1,m2), then Corollary 1 and the §8 residue estimate need to be recomputed; if (52) is confirmed, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 13, used in Corollary 1 to meromorphically continue D_{m1,m2}(s,l) and to extract the residue at s=1 in §8. Lemma 13 claims S = π^2 h^2(-D) N/m2 · σ(l,m1,m2) + O(N^{11/13+ε} m1^{9/13} m2^{-2/13}) for the shifted convolution S = sum_{m2 n2 − m1 n1 = l, n1 ≤ N−1} r(n2)r(n1). The proof line says 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. Section 8 then estimates J2(1,θ) through the residue R of D_{m1,m2}(s,l), and Lemma 14 splits R into four terms Z^{(k,j)}; the two mixed terms are cancelled using G^2(χ_{d1})/d1 + G^2(χ_{d2})/d2 = 0. If the general σ in (52) misses m1- or m2-dependent characters or local densities that appear when m1 ≠ m2, the residue formula and the cancellation in §8 may fail, and with them the key bound J2 ≪ T(θ log T)^{-1/3} and Theorem 1. This is a gap rather than a demonstrated contradiction: the claimed formula may be correct, but the preprint does not prove it for the range of m1,m2 actually used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":59097,"tokens_out":6258,"duration_ms":65099,"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":[{"comment":"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.","section":"§7, Lemma 13"},{"comment":"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.","section":"§3, estimates (11)–(13)"},{"comment":"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.","section":"§4, Lemmas 3 and Theorem 2"}],"minor_comments":[{"comment":"There are numerous typos: 'principle ideal' should be 'principal ideal', 'posess' should be 'possess', and 'the filed' should be 'the field'.","section":"§2"},{"comment":"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.","section":"§5, Lemma 14 statement"},{"comment":"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.","section":"§3, notation of φ(u)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper does not appear circular: the mollifier is standard, no fitted parameters are introduced, and the main theorem is not assumed. The decisive issue is the missing general-case proof of Lemma 13, a gap that is explicitly acknowledged in the text. The second concern about the value-distribution estimates for real characters is also worth resolving. If the author can supply the missing arguments or point to precise statements covering the needed range, the result would be very suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things. First, this is the first unconditional proof that the Epstein zeta-function of a binary positive definite quadratic form, and more generally real linear combinations of Hecke L-functions over imaginary quadratic fields, have a positive proportion of zeros on the critical line. Second, the proof as written is not fully self-contained: the key non-diagonal estimate rests on Lemma 13, whose proof is only given for m1=m2=1 with the general case deferred to [30] plus an unspecified parameter from [17, p.278]. That is a real gap, but it reads like a gap in exposition rather than a demonstrated error.\n\nWhat is genuinely new is the treatment of real Hecke characters. The author correctly identifies why previous work on complex characters does not cover Epstein zeta: the additive problem gains a main term, and the mixed terms in Lemma 14 cancel via the Gauss sum identity. The quantitative form N0(T) >> T log T is what one expects from Selberg's method. The paper is honest about what is imported: value-distribution estimates (11)-(13) are quoted from [28] with only a skeleton, and Lemma 3 is deferred to [25]. For this area that is normal practice, but it does make verification harder.\n\nThe load-bearing soft spot is Lemma 13. The author explicitly states that [30] proves the shifted convolution formula only when m1=m2=1, and says the general coprime case follows by using the precise value of sigma from [17, p.278]. That derivation is not carried out. Corollary 1, the residue computation in Section 8, and the estimate for J2 all depend on it. If the general sigma misses an m1- or m2-dependent factor, the cancellation in Section 8, and with it Theorem 1, would not go through. I do not see a contradiction in the paper, and the claim is plausible, but as submitted the theorem is conditional on an unproved lemma.\n\nThe citation pattern is not a problem: the author leans on her own prior papers, but the cited results are published in refereed venues. No circularity, no fitted parameters.\n\nBottom line: for specialists this is an important claim worth refereeing seriously. The referee should demand the complete proof of Lemma 13 for the range of m1,m2 actually used, and precise statements for the value-distribution lemmas. My own verdict would be conditional, not accept.","headline":"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.","tokens_in":59663,"tokens_out":2455,"would_cite":true,"duration_ms":24711,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Epstein zeta-function","Hecke L-functions","critical line zeros","positive proportion of zeros","linear combination of L-functions","shifted convolution sums","Riemann zeta-function","mollifier method"],"falsifier":"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.","tokens_in":58549,"feed_emoji":"🧮","tokens_out":7294,"duration_ms":72656,"temperature":0.7,"pith_summary":"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.","feed_headline":"Positive proportion of Epstein zeta zeros proven on the critical line","feed_subtitle":"A theorem for Hecke L-function combinations yields N0(T) >> T log T zeros at Re s = 1/2, unconditionally.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the positive-proportion method for the Riemann zeta-function that the proof adapts to linear combinations.","marker":"[34]"},{"why":"Provides the value-distribution and mollifier scheme showing that estimates (8)-(10) imply a positive proportion of zeros on the critical line.","marker":"[37]"},{"why":"Establishes estimates (8)-(10) for Hecke L-functions attached to complex ideal class group characters, which the present paper extends to real characters.","marker":"[28]"},{"why":"Contains the additive-problem asymptotic for the shifted convolution sum in the special case $m_1=m_2=1$, which Lemma 13 generalizes.","marker":"[30]"},{"why":"Supplies the parameter used to modify the proof of the shifted-convolution asymptotic to the general coprime case $(m_1,m_2)\\ne(1,1)$.","marker":"[17]"},{"why":"Provides the estimate for the bound on $\\sigma(l,m_1,m_2)$ and a related treatment of the twisted second moment for quadratic fields.","marker":"[10]"},{"why":"Proves the corresponding positive-proportion result for linear combinations with only complex Hecke characters and supplies auxiliary lemmas used in the present proof.","marker":"[25]"}],"fun_headline_variants":["Epstein zeta zeros: positive proportion on critical line","Proof: Many Epstein zeta zeros on critical line","Positive proportion of Epstein zeros at Re(s)=1/2","Epstein zeta: positive proportion of zeros on critical line","Zeros of Epstein zeta: positive proportion on critical line"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Epstein zeta zeros: positive proportion on critical line","Proof: Many Epstein zeta zeros on critical line","Positive proportion of Epstein zeros at Re(s)=1/2","Epstein zeta: positive proportion of zeros on critical line","Zeros of Epstein zeta: positive proportion on critical line"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000499,"raw_usage":{"total_tokens":2347,"prompt_tokens":750,"completion_tokens":1597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":1514}},"tokens_in":366,"tokens_out":1597,"duration_ms":10921,"temperature":1.0,"reasoning_tokens":1514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:08:17.541778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Selberg, On the zeros of Riemann’s zeta-function, Skr","cited_arxiv_id":null,"evidence_quote":"Supplies the positive-proportion method for the Riemann zeta-function that the proof adapts to linear combinations."},{"cited_title":"Random Matrices and their Applications","cited_arxiv_id":null,"evidence_quote":"Provides the value-distribution and mollifier scheme showing that estimates (8)-(10) imply a positive proportion of zeros on the critical line."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes estimates (8)-(10) for Hecke L-functions attached to complex ideal class group characters, which the present paper extends to real characters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the additive-problem asymptotic for the shifted convolution sum in the special case $m_1=m_2=1$, which Lemma 13 generalizes."},{"cited_title":"Iwaniec, Spacing of zeros of Hecke L-functions and the class number problem, Acta Arith","cited_arxiv_id":null,"evidence_quote":"Supplies the parameter used to modify the proof of the shifted-convolution asymptotic to the general coprime case $(m_1,m_2)\\ne(1,1)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the estimate for the bound on $\\sigma(l,m_1,m_2)$ and a related treatment of the twisted second moment for quadratic fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the corresponding positive-proportion result for linear combinations with only complex Hecke characters and supplies auxiliary lemmas used in the present proof."}],"review_version":1}