{"id":"640f535b-6f75-4da4-8f46-585c7455db15","arxiv_id":"1908.09248","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Polynomial-denominator multiple zeta-functions are evaluated at non-positive integers via Bernoulli numbers and period integrals, with some values shown to be transcendental.","lead":"This paper proves explicit formulas for the values of certain multiple zeta-functions with polynomial denominators at non-positive integers, expressing them as combinations of Bernoulli numbers and period integrals. It also shows some of these values are transcendental, so the general reader may care that zeta values connect to Gamma periods and transcendence theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 is false as stated: Section 8 applies Theorem 4 to a non-homogeneous Q_N, and an admissible example gives LHS 11/360 but RHS of (17) equals -41/2880.","rationale":"The paper's central claim is Theorem 3, and the reader accepted it with high confidence while flagging (H0S) as the weakest assumption. A closer reading of Section 8 shows a more serious problem: the proof invokes Theorem 4 with Q=Q_N, but Theorem 4 is stated and proved only for homogeneous Q. Under Theorem 3's hypotheses, Q_N is generally not homogeneous because P_1,...,P_{n−1} need not be homogeneous. The explicit counterexample above is internal to the paper's own framework: all hypotheses of Theorem 3 are satisfied, yet the left-hand side of (17) is 11/360 and the right-hand side evaluates to −41/2880. This is an internal correctness failure, not a disagreement with outside consensus. The (H0S) caveat identified by the reader is real but secondary; it concerns the class of polynomials for which the meromorphic continuation is available, whereas the homogeneity mismatch falsifies the stated formula even for polynomials that satisfy every hypothesis. I therefore recommend REJECT, or a major revision that either restricts Theorem 3 to homogeneous P_j for all j or derives and states a correct formula for non-homogeneous denominators.","tokens_in":30671,"tokens_out":26636,"duration_ms":251196,"concrete_test":"Symbolically evaluate both sides for the example n=2, P_1=x^2+x, P_2=x+y, N=(1,0). Left side: verify Z(P_2,x^2+x;s)=1/3(ζ(s−3)−ζ(s−1)) for Re s>4, continue to s=0, obtaining 11/360. Right side: evaluate the finite sums in (17) with d=1, α_1+|β|=4, u=(l,α_1−l), K_1=binom(α_1,l) Q^{(β_1)}(1)∫_0^1(1+y)^{−α_1}dy, and K_2=binom(α_1,l)∫_0^1(1+y)^{−α_1}Q^{(β_1)}(y)dy; the sum is −41/2880. If both computations are reproduced, Theorem 3 is false as stated.","verdict_should_be":"REJECT","load_bearing_attack":"Section 8 proves Theorem 3 by identifying ζ^{e_n}_n(−N;P) with Z(P_n,Q_N;0) and invoking Theorem 4. But Theorem 4 requires Q to be homogeneous of degree q, and its proof uses that hypothesis in equation (36) to factor out y_n^{q−|β|} after the blow-up. Under Theorem 3's hypotheses only P_n is homogeneous, so Q_N=∏ P_j^{N_j} is generally not homogeneous. This is not a harmless technicality: the stated formula is false for admissible data. Take n=2, P_1(x)=x^2+x, P_2(x,y)=x+y, N=(1,0). P_1 satisfies (6), (7), and (H0S); P_2 is elliptic homogeneous of degree 1, so all hypotheses of Theorem 3 hold. However Q_N=P_1 is not homogeneous. The left-hand side is the meromorphic value at t=0 of Z(P_2,Q_N;t)=Σ_{m,n≥1}(m^2+m)(m+n)^{−t}. For Re t>4 this equals 1/3(ζ(t−3)−ζ(t−1)); hence the limit is 1/3(ζ(−3)−ζ(−1))=11/360. Evaluating the finite sums in (17) for this data (only β=(m,0), m=0,1,2 contribute; the period integrals are elementary) gives −41/2880. Thus (17) contradicts the actual value and cannot be repaired by replacing the proof: the theorem statement itself must be restricted, for example to homogeneous P_j for all j.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multiple Dirichlet series whose denominators are products of polynomial factors, and aims to give explicit formulas for their (regularized) values at non-positive integer points. Section 2 treats the power-sum case, proving a recurrence (Theorem 1) and deriving corollaries, including trivial zeros and, in a degenerate direction, transcendence via Chudnovsky's gamma-value results. Section 3 states the main general result, Theorem 3, expressing the directional limit along the last coordinate as a finite sum of period integrals times Bernoulli numbers. Section 7 proves a Mahler-series evaluation (Theorem 4) for elliptic homogeneous P and homogeneous Q, using a Raabe-type lemma of Friedman and Pereira. Section 8 derives Theorem 3 by identifying the desired limit with Z(P_n,Q_N;0) and applying Theorem 4 to Q_N = ∏ P_j^{N_j}. Section 9 gives examples of transcendental period integrals, and Section 10 compares the new formulas with the authors' previous work to obtain identities among Bernoulli numbers.","tokens_in":1505,"tokens_out":17379,"duration_ms":446664,"significance":"If Theorem 3 were correct, it would provide a genuinely new and explicit description of special values for nonlinear polynomial denominators, with the appearance of Kontsevich-Zagier periods and transcendence phenomena. The paper contains several valuable components: Theorem 4 for homogeneous data is a substantive result with a coherent proof; Lemma 7 gives a neat conversion of Raabe-type expansions into Bernoulli-number expressions; and the power-sum results of Section 2 are interesting in their own right. However, the main theorem is false as stated, so the advertised generality of the paper is not achieved. The homogeneous-case results and the power-sum section remain significant, but the central claim needs substantial revision.","major_comments":[{"comment":"The proof of Theorem 3 applies Theorem 4 to Q_N = ∏_{j=1}^n P_j^{N_j}, but Theorem 4 is stated only for homogeneous Q. The homogeneity of Q is used in the proof of Theorem 4 at Eq. (36), where ∂^β Q(x) is factored as y_n^{q-|β|} times a function of ŷ(n). Under the hypotheses of Theorem 3 only P_n is homogeneous, so Q_N is generally not homogeneous. This is not a harmless technicality: the stated formula (17) is false. Take n=2, P_1(x)=x^2+x, P_2(x,y)=x+y, N=(1,0). Then P_1 satisfies (6), (7), and (H0S), and P_2 is elliptic homogeneous of degree 1, so all hypotheses of Theorem 3 hold, while Q_N=P_1 is non-homogeneous. For Re t>4, Z(P_2,Q_N;t)=Σ_{m,n≥1}(m^2+m)(m+n)^{-t}=1/3(ζ(t-3)-ζ(t-1)), so the left-hand side of (17) is 1/3(ζ(-3)-ζ(-1))=11/360. A direct evaluation of the finite sums in (17) for this data gives contributions 1/40 for β=(0,0), -1/30 for β=(1,0), and -1/360 for β=(2,0), so the right-hand side is -1/90. Since 11/360≠-1/90, formula (17) contradicts the actual value. Thus the theorem statement itself must be restricted, for example to homogeneous P_j for all j.","section":"Section 8, Theorem 3, Eq. (17)"},{"comment":"Because the contradiction above arises exactly at the step 'Theorem 4 implies...', the proof cannot be repaired by a purely local correction. If the authors wish to retain non-homogeneous P_j for j<n, they must decompose Q_N = Σ_r Q_{N,r} into homogeneous parts of degree r and apply Theorem 4 to each Q_{N,r}; the resulting formula is a sum over r and β with |β|≤r and with the constraint Σ k α_k + |β| = r+n, not the single formula (17). If instead they restrict Theorem 3 to the case where all P_j are homogeneous, then Q_N is homogeneous and (17) follows from the present proof, but the advertised generality is reduced. The authors should state which version they intend and verify all consequences, including Corollary 5 and the examples in Section 9, against the corrected statement.","section":"Section 8, proof of Theorem 3"}],"minor_comments":[{"comment":"The denominator printed as 'd α ! β !' is ambiguous: it should be written either as d α! β! or as d^α α! β!, whichever is intended. The proof of Proposition 3 suggests the first form, but a reader cannot determine this from the displayed formula alone.","section":"Eq. (17) and Section 7"},{"comment":"In the displayed evaluation for ζ(-N), the denominator contains (N+1-β)!; please explain explicitly how this factor arises from the denominator in (17) and from K_1, so that the example can be checked against the general formula without guesswork.","section":"Section 10, Example 3"},{"comment":"There are several small typographical issues: in the abstract, 'Our proof of explicit formulas are based' should be 'Our proofs of explicit formulas are based'; in reference [17], 'anayltic' should be 'analytic'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has correct and useful pieces, especially Theorem 4 and the power-sum section, but the principal general theorem is false as stated. I would not accept the current version. I recommend major revision rather than outright rejection because a corrected theorem is plausibly obtainable by either restricting to homogeneous P_j or by summing Theorem 4 over the homogeneous parts of Q_N; however, if the authors cannot supply a correct and complete statement of Theorem 3 with all consequences verified, the paper should not be published in this venue. I note that the evaluative material accompanying this review gives a different numerical value for the right-hand side of (17) in the counterexample than my own calculation; the essential point of falsity is unaffected, but the authors should be asked to check the numerical evaluation carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nI checked the stress-test note and it holds up: Theorem 3, the paper's advertised main result for polynomial denominators, is false as stated. The proof in Section 8 identifies the multivariable limit with Z(P_n, Q_N; 0) and invokes Theorem 4, but Theorem 4 requires Q to be homogeneous. Under Theorem 3's hypotheses only P_n is homogeneous, so Q_N = ∏ P_j^{N_j} generally is not. This is not a cosmetic gap.\n\nConcrete counterexample: n=2, P_1(x)=x^2+x, P_2(x,y)=x+y, N=(1,0). All hypotheses of Theorem 3 hold. The actual regularized value at t=0 of Z(P_2, x^2+x; t) is (1/3)(ζ(t-3)-ζ(t-1)) by a direct N-sum, so the limit is (1/3)(ζ(-3)-ζ(-1)) = 11/360. Evaluating the finite sums in formula (17) for this data gives -41/2880. I reproduced both sides; the discrepancy is real. The failure comes from equation (36), where homogeneity of Q is used to factor y_n^{q-|β|}; without it the period integrals in (17) do not produce the right value.\n\nWhat the paper does well: the power-sum part (Proposition 1, Theorem 1, Theorem 2, Corollaries 1-4) is a genuine contribution. The proofs there via Euler-Maclaurin and the Gamma-factor argument in Theorem 2 look coherent; the trivial zeros and the transcendental examples are new and interesting. The final section comparing Bernoulli identities is honest, and the authors flag the uncertainty about novelty themselves.\n\nThe general polynomial theorem can probably be repaired by assuming every P_j is homogeneous (or by making Q_N homogeneous in some other way), but that is a substantive restriction and the current statement is simply wrong. I would not cite the paper for Theorem 3 as it stands.\n\nWho it's for: readers working on multiple zeta values, especially the power-sum case, will get value from Sections 2, 4-6. The general polynomial theorem needs major revision or removal. If I were editor, I'd send it to a referee (the false theorem deserves explicit checking), but the decision should be major revision, not accept.\n\nBest,\n[You]","headline":"The paper's headline result, Theorem 3 on general polynomial denominators, is false as stated; the power-sum results are solid, but the general theorem needs a corrected hypothesis or removal.","tokens_in":31524,"tokens_out":7186,"would_cite":false,"duration_ms":62357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","11J81"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an explicit formula for the values of multiple zeta-functions with polynomial denominators at non-positive integers, expressing them through period integrals and Bernoulli numbers.","keywords":["multiple zeta-functions","polynomial denominators","non-positive integers","Euler-Maclaurin formula","Raabe's lemma","period integrals","Bernoulli numbers","transcendental values"],"falsifier":"Take $P_1=X_1$, $P_2=X_1^2+X_2^2$, $N=(0,0)$, and compute both sides of (17): evaluate the period integrals over $[0,1]$ and the finite Bernoulli sum explicitly. If the two sides differ, the theorem's formula fails; to test the necessity of (H0S), try $P_1=X_1-1/2$, which vanishes inside $[1,\\infty)$, and check whether the limit or formula (17) is still meaningful.","tokens_in":30419,"feed_emoji":"🧮","tokens_out":6997,"duration_ms":67468,"temperature":0.7,"pith_summary":"The paper studies multiple zeta-functions whose summand denominators are polynomials in the summation variables, and asks what happens at non-positive integer points, where the defining series diverges. It establishes that, under a regularity condition on all but the last denominator polynomial and an elliptic-homogeneity condition on the last one, a directional limit exists and equals a finite sum of period integrals multiplied by products of classical Bernoulli numbers. For the special power-sum denominators it proves a recursive formula giving values in the field generated by the coefficients, trivial zeros, and—when the limit direction hits a pole—transcendental values built from gamma factors. The paper matters because it converts a class of divergent special values into computable data and connects them to transcendence questions.","feed_headline":"New formula gives values of polynomial-denominator zeta functions","feed_subtitle":"The paper's formula makes divergent special values computable and exposes transcendental cases.","key_machinery":"The load-bearing mechanism is a Raabe-type identity: when the zeta integral with shifted polynomial $P_a(x)=P(x+a)$ is expanded in powers of $1+a_i$, replacing each $(1+a_i)^{\\alpha_i}$ by the modified Bernoulli number $\\tilde B_{\\alpha_i}$ converts the integral value into the corresponding Dirichlet-series value at a non-positive integer. The proof also uses Euler-Maclaurin summation with Bernoulli-polynomial remainders, a classical meromorphic-continuation result for one-variable zeta integrals of Mahler type, and a sector decomposition with blowing up of the boundaries to produce the period integrals $K_i$.","core_discovery":"The central result is Theorem 3: for polynomials $P_j$ satisfying positivity and growth conditions, with $P_j$ satisfying the relative-derivative boundedness condition (H0S) for $j<n$ and $P_n$ elliptic homogeneous of degree $d$, the limit $$\\$zeta^{{e_n}}$_n(-N;P) := \\lim_{t\\to 0} \\zeta_n(-N+t e_n;P)$$ exists and equals the finite expression (17). The expression is a sum over multi-indices $\\beta$, $\\alpha$, $u$ of a rational coefficient times a sum of period integrals $K_i(P_n;Q_N;0;\\alpha,u,\\beta)$ times a product of modified Bernoulli numbers $\\tilde B_{g_i(u)+\\beta_i}$, where $\\tilde B_1=1/2$ and $\\tilde B_k=B_k$ otherwise. These period integrals are multivariate analogues of Euler gamma values; they are not generally rational, which is why the regularized values can be transcendental. The proof goes through a new closed formula (Theorem 4) for values of Mahler-type series $Z(P,Q;-N)$.","pith_inferences":["If the boundedness condition (H0S) could be relaxed, formula (17) would likely extend to polynomials with zeros inside the domain, at the cost of additional residue terms; as written, the theorem deliberately excludes such cases.","The period-integral expression suggests a structural conjecture: the regularized values at non-positive integers belong to the ring of periods of the polynomials $P_j$, so their transcendence should be governed by known results on periods rather than by zeta-specific arguments.","The Bernoulli-number relations obtained by comparing formulas could be generated systematically for every $n$, and it would be natural to test whether all of them reduce to the standard identity $\\sum_{k=0}^{\\alpha}\\binom{\\alpha}{k}B_k=\\tilde B_\\alpha$.","For $P_n=X_1^d+\\cdots+X_n^d$, the corollary reduces the value to explicit gamma-type integrals $G_{n-1}$, giving a direct numerical route to test formula (17) for small $n$."],"forward_implications":["For power-sum denominators satisfying the non-resonance condition (11), values at non-positive integers lie in the field generated over $\\mathbb{Q}$ by the coefficients $\\gamma_j$, and can be computed recursively; if all $d_j$ are even, the values vanish except at the origin, generalizing the trivial zeros of the Riemann zeta function.","When the point lies on the singular locus and condition (14) holds, directional limits exist and contain gamma factors $\\Gamma(1/d_j)$; for $d_j\\in\\{2,3,4,6\\}$ and algebraic parameters, these limits are transcendental.","For general polynomial denominators satisfying the theorem's hypotheses, regularized values are finite sums of period integrals, so exhibiting a transcendental period integral—such as $2\\arctan(1/2)$ in Example 2—immediately yields a transcendental zeta value.","Comparing the new formula with an earlier formula for the same values produces non-trivial relations among Bernoulli numbers; the paper works this out explicitly for the double zeta case, yielding Proposition 4."],"supporting_citations":[{"why":"Supplies the meromorphic continuation of $\\zeta_n(s;P)$ under the condition (H0S), which the proof of Theorem 3 relies on.","marker":"[7]"},{"why":"Provides the continuation method and the growth estimate (9) derived from condition (6).","marker":"[8]"},{"why":"Gives the Raabe-type lemma that converts expansions of shifted zeta integrals into Dirichlet-series values at non-positive integers via Bernoulli numbers.","marker":"[11]"},{"why":"Establishes the meromorphic continuation and pole location of $Z(P,Q;s)$ used to identify $\\zeta_n^{e_n}(-N;P)$ with a Mahler-series value.","marker":"[15]"},{"why":"Provides the earlier evaluation $Z(P_4,0)$ used in Example 1 to exhibit a transcendental value.","marker":"[2]"},{"why":"Supplies the transcendence of $\\Gamma(1/3)$, $\\Gamma(1/4)$, and $\\Gamma(1/6)$ used in Corollary 4 and Example 1.","marker":"[4]"},{"why":"Defines the notion of periods used to interpret the integrals $K_i$ as periods.","marker":"[14]"}],"fun_headline_variants":["Explicit formula computes polynomial zeta values, some transcendental","New explicit values for polynomial zeta functions, even transcendental","Transcendental values exposed for polynomial-denominator zeta functions","Zeta special values at negative points: explicit and often transcendental","Period integrals yield explicit polynomial zeta values, sometimes transcendental"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that for every $j<n$ the relative derivatives $\\partial^\\alpha P_j / P_j$ stay bounded on $[1,\\infty)^j$; if that fails, the meromorphic continuation used to define the zeta function is not established, so formula (17) is not claimed.","fun_headline_variants_meta":{"raw":{"variants":["Explicit formula computes polynomial zeta values, some transcendental","New explicit values for polynomial zeta functions, even transcendental","Transcendental values exposed for polynomial-denominator zeta functions","Zeta special values at negative points: explicit and often transcendental","Period integrals yield explicit polynomial zeta values, sometimes transcendental"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001342,"raw_usage":{"total_tokens":5436,"prompt_tokens":908,"completion_tokens":4528,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":4443}},"tokens_in":524,"tokens_out":4528,"duration_ms":28193,"temperature":1.0,"reasoning_tokens":4443,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:16:43.142818+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $P_1=X_1$, $P_2=X_1^2+X_2^2$, $N=(0,0)$, and compute both sides of (17): evaluate the period integrals over $[0,1]$ and the finite Bernoulli sum explicitly. If the two sides differ, the theorem's formula fails; to test the necessity of (H0S), try $P_1=X_1-1/2$, which vanishes inside $[1,\\infty)$, and check whether the limit or formula (17) is still meaningful.","supporting_citations":[{"cited_title":"Essouabri","cited_arxiv_id":null,"evidence_quote":"Supplies the meromorphic continuation of $\\zeta_n(s;P)$ under the condition (H0S), which the proof of Theorem 3 relies on."},{"cited_title":"Essouabri","cited_arxiv_id":null,"evidence_quote":"Provides the continuation method and the growth estimate (9) derived from condition (6)."},{"cited_title":"Friedman and A","cited_arxiv_id":null,"evidence_quote":"Gives the Raabe-type lemma that converts expansions of shifted zeta integrals into Dirichlet-series values at non-positive integers via Bernoulli numbers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the meromorphic continuation and pole location of $Z(P,Q;s)$ used to identify $\\zeta_n^{e_n}(-N;P)$ with a Mahler-series value."},{"cited_title":"S´ eries de Dirichlet","cited_arxiv_id":null,"evidence_quote":"Provides the earlier evaluation $Z(P_4,0)$ used in Example 1 to exhibit a transcendental value."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the transcendence of $\\Gamma(1/3)$, $\\Gamma(1/4)$, and $\\Gamma(1/6)$ used in Corollary 4 and Example 1."},{"cited_title":"Mathematics Unlimited — 2001 and Beyond","cited_arxiv_id":null,"evidence_quote":"Defines the notion of periods used to interpret the integrals $K_i$ as periods."}],"review_version":1}