{"id":"9d5fe3af-0de0-480d-a627-006cb8b53c46","arxiv_id":"2411.08863","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Random variables whose moments equal the completed Dedekind zeta functions of Q, Q(i), and Q(sqrt(-2)) imply the first two Li coefficients of these zeta functions are positive.","lead":"This paper constructs random variables whose moments are the completed Dedekind zeta functions of the rationals and two imaginary quadratic fields, then uses them to prove the first two Li coefficients are positive for these fields. It extends the probabilistic interpretation of the Riemann zeta function from Biane, Pitman, and Yor to two new number fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.2's Poisson summation identity for the imaginary quadratic case has the wrong scale factor (|u| instead of |u|^2/(2√d)); this invalidates the proof as written, though the nonnegativity conclusion appears repairable.","rationale":"The reader's weakest-assumption analysis identifies exactly the step I consider most load-bearing: the Poisson summation formula in Lemma 2.2 for the imaginary quadratic lattices. My independent derivation agrees that the scale factor is incorrect. The displayed identity has |u|, whereas two-dimensional Poisson summation with the paper's self-dual measure and character gives |u|^2/(2√d). This is not a mere typo in a peripheral computation: Lemma 2.2 supplies the nonnegativity of ψ(t), and Theorem 1.1 depends on ψ being a probability density. As written, the proof of the main theorem therefore has a genuine gap. The gap is nevertheless repairable. The corrected factor is positive, and the argument's positivity mechanism — the lower bound on |uℓ*|^2 forcing each summand f∞(uℓ*) to be nonnegative — is unaffected by which positive factor appears. Thus the conclusion of Lemma 2.2 is very likely true, and the central claim should survive revision. I also note a smaller unproved step in Lemma 2.1, where self-duality of f∞ is inferred from equality of zeta integrals via 'uniqueness of the zeta integral'; this is defensible for radial Schwartz functions but should be stated. The cumulant proof of Proposition 2.1 is clean and independent of the disputed constant, so no further load-bearing issue emerged. Because the identified error requires correction but is not fatal, the reader's CONDITIONAL verdict remains appropriate; I do not move the verdict.","tokens_in":7022,"tokens_out":26203,"duration_ms":221863,"concrete_test":"Compute directly the d = 1 case of the displayed Poisson identity in Lemma 2.2 at x∞ = 1/3. Using the paper's f∞(z) = 4π|z|^2(π|z|^2 - 1)e^{-2π|z|^2}, evaluate the left side summed over Gaussian integers m + ni with |m|,|n| ≤ N, and the right side summed over the dual lattice (1/2)(Z⊕iZ), both with the stated factor |u| = 3. For large N the identity should fail; replacing the factor by |u|^2/(2√d) = 9/2 should make the two sides agree within truncation error. This settles whether the Poisson constant is wrong. If it is, patch Lemma 2.2 with the corrected factor and re-verify the positivity bound π|uℓ*|^2 - 1 ≥ π^2/(4d) - 1 > 0 for d = 1,2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the Poisson summation identity in Lemma 2.2 for K = Q(√-d), d = 1,2. The paper displays: sum_{ℓ∈O_K\\{0}} f∞(x∞ℓ) = |u| sum_{ℓ*∈O_K^*\\{0}} f∞(uℓ*), where u = x∞^{-1} and O_K^* = (2√-d)^{-1}(Z⊕√-dZ). With the paper's self-dual Haar measure on C (twice Lebesgue measure) and the additive character under which e^{-2π|z|^2} is self-dual, the correct two-dimensional Poisson summation factor is |u|^2 / μ(C/O_K) = |u|^2/(2√d), not |u|. The lattice O_K has covolume 2√d in this normalization, and the scaling of a Schwartz function in two variables contributes |u|^2. Thus the displayed equality is false as written. This matters because Lemma 2.2 is the only source of nonnegativity of the density ψ(t) in Theorem 1.1; without it, the random variable is not constructed. However, the error is repairable: the missing factor is positive, and the positivity of the summands on the right is independent of that factor, since the lower bound |uℓ*|^2 ≥ π/(4d) still gives π|uℓ*|^2 - 1 ≥ π^2/(4d) - 1 > 0 for d = 1,2. A secondary gap is the appeal to 'uniqueness of the zeta integral' in Lemma 2.1 to conclude f∞ is self-dual; this is plausible for radial functions but is not proved. The central probabilistic construction and the cumulant derivation of λ1, λ2 > 0 otherwise appear sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for K = Q, Q(√-1), and Q(√-2), a random variable X on (0,∞) whose Mellin transform equals |D|^{-s/2}ξ_K(s) for every s ∈ C, where ξ_K is the completed Dedekind xi function. The construction uses an adelic Schwartz-Bruhat function whose global zeta integral is s(s-1)Z_K(s), and the density is obtained by fibering the idelic norm over the idele class group. The author then uses elementary cumulant identities to prove that the first two Li coefficients for these zeta functions are positive, extending the Biane–Pitman–Yor result for the Riemann zeta function.","tokens_in":7371,"tokens_out":18757,"duration_ms":143507,"significance":"If the construction is made rigorous, this is a valuable contribution: it gives a genuine probabilistic interpretation of Dedekind zeta functions for three explicit number fields and derives the positivity of the first two Li coefficients from moment/cumulant considerations rather than from numerical evaluation. The adelic framework is natural, the choice of f∞ is explicit, and the cumulant argument is clean and free of circularity. The paper also openly discusses the obstruction to generalizing the argument to other imaginary quadratic fields, which is helpful. However, the main lemma establishing nonnegativity of the density contains a false factor in a Poisson summation identity, and the proof of self-duality of f∞ is incomplete; both points must be repaired before the central theorem is fully established.","major_comments":[{"comment":"The displayed Poisson summation identity in the imaginary quadratic case is incorrect. The paper states ∑_{ℓ∈O_K\\{0}} f∞(x∞ℓ) = |u| ∑_{ℓ*∈O_K^*\\{0}} f∞(uℓ*) with u = x∞^{-1}, but with the paper's self-dual measure on C (twice Lebesgue measure) and the character under which e^{-2π|z|^2} is self-dual, the correct two-dimensional identity is ∑_{ℓ∈O_K\\{0}} f∞(x∞ℓ) = (|u|^2/(2√d)) ∑_{ℓ*∈O_K^*\\{0}} f∞(uℓ*). The factor |u| is the one-dimensional scaling factor; in two dimensions the dilation contributes |u|^2 and the lattice covolume contributes 1/(2√d). As written the equality is false, and since Lemma 2.2 is the only source of nonnegativity of ψ(t), the proof of Theorem 1.1 is incomplete. The error is repairable: the missing factor is positive, and the subsequent lower-bound argument π|uℓ*|^2 - 1 ≥ π^2/(4d) - 1 > 0 for d = 1,2 is unaffected, so the nonnegativity conclusion can be restored.","section":"§2.2, Lemma 2.2"}],"minor_comments":[{"comment":"In the real case of the proof, the expression 'g′_1(s)' should be 'g′_1(x)'.","section":"§2.1, Lemma 2.1"},{"comment":"The reduction 'we can suppose without loss of generality' for the fractional ideal M should be made more explicit: since K has class number 1, M = αO_K, and one should absorb α into the arbitrary value of x∞; the two cases in the argument should then be conditioned on |x∞α|^2 rather than on |x∞|^2 alone.","section":"§2.2, Lemma 2.2"},{"comment":"The additive character ψ_v and the self-dual Haar measure are not specified in detail for the complex place. For completeness, the author should state that ψ_∞(z) = e^{-4πi Re(z)} and dμ_∞ = 2 dx dy, which makes the self-duality of e^{-2π|z|^2} and the Poisson summation constant in Lemma 2.2 directly verifiable.","section":"§1.3, §2.1"},{"comment":"The notation O_K^* for the dual lattice is easily confused with the unit group O_K^×; consider using O_K^∨ or another symbol.","section":"§2.2, Lemma 2.2"},{"comment":"The statement that log X and log |Y| have the same moment generating functions should explicitly mention that they agree on an open neighborhood of 0, which is what justifies equality of distributions.","section":"§2.2, Remark after Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is well motivated and the overall strategy is sound, but the false factor in Lemma 2.2 is a genuine error in a load-bearing proof step. It is clearly repairable, and I do not see grounds for rejection. The secondary gap concerning the 'uniqueness of the zeta integral' in Lemma 2.1 should also be addressed, as it underlies the self-duality used in the same Poisson summation argument. I recommend major revision rather than minor, because the manuscript as written does not yet rigorously establish the nonnegativity of the density."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a genuine, mostly sound extension of Biane–Pitman–Yor's probabilistic zeta construction to Q(i) and Q(√−2), with one repairable scaling error in the imaginary-quadratic Poisson summation step. I agree with the reader's conditional verdict.\n\nThe new thing is the construction itself: a random variable X for each of Q, Q(i), Q(√−2) whose complex moments are |D|^{-s/2} ξ_K(s). The density is built from an adelic zeta integral with a Schwartz–Bruhat function chosen so that the Mellin transform is s(s−1)Z_K(s). That is a legitimate and elegant extension, not present in the cited BPY paper. The cumulant argument for the Li coefficients is clean and correct: λ1 comes from Jensen, λ2 from variance, and the strict positivity follows.\n\nThe soft spot is Lemma 2.2. For K=Q(i) or Q(√−2), the displayed Poisson summation identity is wrong. When you scale a function on C by u, the two-dimensional Poisson formula carries a factor |u|^2, and the lattice O_K has covolume 2√d in the paper's self-dual measure, so the leading constant should be |u|^2/(2√d), not |u|. The stress-test note is right. The good news is that the error is not load-bearing for the conclusion: the correct constant is positive, and the estimate π|uℓ*|^2−1 ≥ π^2/(4d)−1 > 0 for d=1,2 still holds, so the sum is nonnegative. The proof can be repaired by changing one factor. This means the central claim is true but the written proof is invalid at that point.\n\nA smaller issue: Lemma 2.1's inference of self-duality from equality of zeta integrals is stated as ‘uniqueness of the zeta integral’ without saying that the functions involved are radial. With the radiality assumption it is fine, but the sentence is too quick.\n\nOverall: the mathematics is basically sound, the error is repairable, and the exposition of the adelic background is clear. The immediate consequence, positivity of the first two Li coefficients, is modest, but the probabilistic representation of ξ_K for these fields is a useful tool. This deserves a serious referee. I would send it out and ask the referee to verify the Poisson constant and the self-duality proof.","headline":"A genuine extension of Biane–Pitman–Yor to two imaginary quadratic fields, with a repairable Poisson-summation scaling error in Lemma 2.2; deserves review after a small fix.","tokens_in":7924,"tokens_out":9804,"would_cite":true,"duration_ms":82041,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R42","11M06","60E10","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Q, Q(√-1), and Q(√-2), a random variable X exists whose complex moments equal the Dedekind xi function, and this proves the first two Li coefficients are positive.","keywords":["Dedekind zeta function","Li coefficients","zeta integrals","random variables","adelic Poisson summation","imaginary quadratic fields","Mellin transform","probabilistic number theory"],"falsifier":"Take $K = \\mathbb{Q}(\\sqrt{-1})$, choose a concrete $x \\in A^\\times$ (for instance $x_\\infty = 1$ and $x_p = 1$ for all primes $p$), and numerically evaluate the identity in Lemma 2.2, comparing the left-hand sum over $\\mathcal{O}_K \\setminus \\{0\\}$ with the right-hand dual sum using the correct lattice covolume $2\\sqrt{d}$. If the two sides differ, the nonnegativity proof fails.","tokens_in":6784,"feed_emoji":"🎲","tokens_out":7252,"duration_ms":57095,"temperature":0.7,"pith_summary":"This paper aims to give a probabilistic interpretation of the Dedekind zeta functions of three number fields: the rationals and the two imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-1})$ and $\\mathbb{Q}(\\sqrt{-2})$. Its main theorem produces, for each of these fields, a random variable $X$ whose complex moments $E(X^s)$ equal the normalized Dedekind xi function $|D|^{-s/2}\\xi_K(s)$. The existence of such an $X$ is used to prove that the first two Li coefficients of these zeta functions are positive, a condition connected to the Riemann hypothesis via Li's criterion. This extends a 2001 probabilistic treatment of the Riemann zeta function to a small family of number fields.","feed_headline":"Random variable moments equal Dedekind xi for three fields","feed_subtitle":"This proves the first two Li coefficients are positive, one step toward the Riemann hypothesis for these fields.","key_machinery":"The key object is the adelic zeta integral $Z(f,s) = \\int_{A^\\times} |x|^s f(x) \\, d^\\times x$, together with a deliberately chosen Schwartz–Bruhat function $f$ (a test function that is smooth and rapidly decreasing on archimedean components and locally constant and compactly supported on nonarchimedean ones). Its archimedean component $f_\\infty$ is engineered, as a derivative of a Gaussian, so that the local zeta integral equals $s(s-1)$ times the appropriate Gamma factor. The argument then invokes Poisson summation over the lattice of algebraic integers to show the resulting density on the idele class group is nonnegative, and uses self-duality $\\hat{f} = f$ to identify the density that generates the xi function.","core_discovery":"The central discovery is that the Dedekind xi function of $K$ can be realized exactly as the moment-generating function of a probability distribution on the positive reals. The construction works by choosing a Schwartz–Bruhat function $f$ on the adeles of $K$ whose global zeta integral equals $s(s-1)Z_K(s)$, then showing that the function $c_K^{-1} t^{-1} \\int_{A_t^\\times} f \\, d^\\times x$ is a nonnegative density integrating to 1. The paper then reads $\\lambda_1$ and $\\lambda_2$ as cumulants of the random variable $L = -\\log X$, proving their positivity.","pith_inferences":["If the scaling-factor issue in the Poisson summation step is repaired, the same construction plausibly extends to other imaginary quadratic fields of class number 1; the paper's remark suggests the summands would no longer all be positive for discriminants $d \\ge 3$, but a more delicate positivity argument might still hold.","The cumulant interpretation of the Li coefficients suggests that higher Li coefficients could be studied through the moment-generating function of $-\\log X$ without needing the full density; for instance, $\\lambda_3$ would involve the third cumulant.","The probabilistic framework could enable numerical sampling of $X$ to estimate Dedekind zeta values in the critical strip, provided the density is efficiently simulatable.","A testable extension is to check whether the density for $K = \\mathbb{Q}(\\sqrt{-3})$ becomes negative when the correct lattice covolume is used; the paper's remark indicates this is where the argument would break down."],"forward_implications":["The first two Li coefficients $\\lambda_1$ and $\\lambda_2$ are positive for $\\mathbb{Q}$, $\\mathbb{Q}(\\sqrt{-1})$, and $\\mathbb{Q}(\\sqrt{-2})$, a necessary condition for all nontrivial zeros of these Dedekind zeta functions to lie on the critical line.","The random variable $X$ gives a probabilistic model of the Dedekind xi function, so the functional equation $\\xi_K(s) = \\xi_K(1-s)$ is mirrored by a symmetry of the moment function.","The distribution of $X$ lives on the idele class group $A^\\times / K^\\times$, meaning analytic questions about the zeta function become questions about this probability measure.","By Li's criterion, if the construction could be extended to produce positive $\\lambda_n$ for all $n$, the nontrivial zeros of the corresponding Dedekind zeta function would all lie on the critical line."],"supporting_citations":[{"why":"Supplies the original method of interpreting a Mellin transform of a zeta-type function as the moment of a random variable, which this paper extends to Dedekind zeta functions.","marker":"[1]"},{"why":"Defines the Li coefficients and states the theorem that positivity of all Li coefficients implies all nontrivial zeros lie on the critical line, providing the motivation for Proposition 2.1.","marker":"[2]"},{"why":"Provides the algebraic number theory background (discriminants, gamma factors, class number, places) used throughout the construction.","marker":"[3]"},{"why":"Supplies the Tate zeta integral machinery, including the functional equation and the local measure conventions that underlie the Poisson summation identity in Lemma 2.2.","marker":"[4]"}],"fun_headline_variants":["Zeta integrals encode probability laws for three fields","Dedekind xi as moment-generating function proves Li positivity","Probabilistic twist on zeta: moments yield positive Li coefficients","Three zeta functions, one probability law: Li coefficients proven positive","From zeta integrals to probability: new proof of Li positivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction leans on a Poisson summation step over the lattice of algebraic integers; if the scaling factor in that step does not match the true covolume of the lattice, the density might not be nonnegative.","fun_headline_variants_meta":{"raw":{"variants":["Zeta integrals encode probability laws for three fields","Dedekind xi as moment-generating function proves Li positivity","Probabilistic twist on zeta: moments yield positive Li coefficients","Three zeta functions, one probability law: Li coefficients proven positive","From zeta integrals to probability: new proof of Li positivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000666,"raw_usage":{"total_tokens":2934,"prompt_tokens":731,"completion_tokens":2203,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":347,"completion_tokens_details":{"reasoning_tokens":2117}},"tokens_in":347,"tokens_out":2203,"duration_ms":14362,"temperature":1.0,"reasoning_tokens":2117,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:18:30.580987+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $K = \\mathbb{Q}(\\sqrt{-1})$, choose a concrete $x \\in A^\\times$ (for instance $x_\\infty = 1$ and $x_p = 1$ for all primes $p$), and numerically evaluate the identity in Lemma 2.2, comparing the left-hand sum over $\\mathcal{O}_K \\setminus \\{0\\}$ with the right-hand dual sum using the correct lattice covolume $2\\sqrt{d}$. If the two sides differ, the nonnegativity proof fails.","supporting_citations":[{"cited_title":"Biane, J","cited_arxiv_id":null,"evidence_quote":"Supplies the original method of interpreting a Mellin transform of a zeta-type function as the moment of a random variable, which this paper extends to Dedekind zeta functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Li coefficients and states the theorem that positivity of all Li coefficients implies all nontrivial zeros lie on the critical line, providing the motivation for Proposition 2.1."},{"cited_title":"Neukirch, Algebraic Number Theory","cited_arxiv_id":null,"evidence_quote":"Provides the algebraic number theory background (discriminants, gamma factors, class number, places) used throughout the construction."},{"cited_title":"Fourier analysis in number fields, and Hecke’ s zeta-functions","cited_arxiv_id":null,"evidence_quote":"Supplies the Tate zeta integral machinery, including the functional equation and the local measure conventions that underlie the Poisson summation identity in Lemma 2.2."}],"review_version":1}