{"id":"71c00c35-9bd0-4b89-bbb8-e14a715dd3eb","arxiv_id":"2506.20150","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For partially twisted multiple zeta-functions with polynomial denominators, the paper proves explicit formulas for the special values at non-positive integers and exhibits transcendental values in examples.","lead":"This mathematics paper works out exact values, at non-positive integer points, of a family of infinite sums called partially twisted multiple zeta-functions, where the denominators are polynomials. It extends a known fully twisted case and earlier linear and power-sum cases, and it shows that some of these exact values are transcendental numbers such as powers of pi.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the Mellin-Barnes/residue reduction is internally consistent and Proposition 6 supplies the needed moderate-growth input.","rationale":"The reader's weakest-assumption identification matches my own reading: the genuinely load-bearing technical premise is Proposition 6, the moderate-growth extension of de Crisenoy's theorem, proved through Proposition 15. I examined the places where this premise is used — repeated contour shifts in Propositions 31-33, the definition of F in Definition 21, and the final evaluation at non-positive integers in Theorem 24 — and found the logic consistent. The residue computations for the special cases d=1 and d=2 agree with the general formula when specialized, which is a good internal cross-check. I also checked potential edge cases: the a0=0 convention, the d=1 limit of Theorem 24 against Theorem 17, the positivity of g(s_T;-ell) in the second sum of (6.2), and the appearance of positive integer arguments in fully twisted values. The latter is explicitly acknowledged by the authors and does not invalidate the theorem, since it states a reduction to de Crisenoy values rather than a fully elementary closed form. The only part I would want verified independently is the uniformity claim in Proposition 15, but I found no concrete error and do not regard it as a demonstrated flaw. Accordingly, the reader's ACCEPT verdict is unchanged.","tokens_in":29125,"tokens_out":28473,"duration_ms":311272,"concrete_test":"Independently re-derive the induction in Proposition 15, tracking the number of terms and the degree of the polynomial coefficients in s appearing at each stage of de Crisenoy's recurrence, and verify that the implicit constant in (3.4) is independent of the choice of sequence (f_n) for f in B(1,M); then confirm that the choice of eps in Section 3.3 converts the factor exp(O(eps) sum |Im s_t|) into the claimed polynomial bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central value formula (6.2) rests on two analytic inputs: de Crisenoy's entirely theorem (Proposition 3) and the new moderate-growth estimate for the fully twisted zeta function (Proposition 6). Proposition 6 is proved in Section 3 via the uniform estimate in Proposition 15. The proof of Proposition 15 is compressed where it inherits de Crisenoy's induction and asserts uniform constants for f_k in B(1,M), but the mechanism there — integration by parts against uniformly bounded primitives — is exactly what is needed, and the explicit power m = floor((q+aTp+2)/eps0)+1 in (3.3)-(3.4) accounts for the required decay. No missing residue, false limit, or unsupported interchange was found in the contour shifts of Sections 4, 5, and 7. The paper also candidly states that the resulting formula can involve fully twisted values at positive integer arguments, which limits 'explicitness' but is not a correctness gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the partially twisted multiple zeta-function ζ_n(s;P;µ_{n-1}) with general polynomial denominators, in the case where all but one of the summation variables carry twists. Using the multiple Mellin-Barnes formula, the authors reduce the problem to the fully twisted case solved by de Crisenoy. The main results are Theorem 22 and Corollary 23, giving meromorphic continuation to C^T and a description of the singular hyperplanes, and Theorem 24, which gives an explicit formula for the values at non-positive integer points in terms of de Crisenoy's fully twisted values, twisted Bernoulli numbers, and multinomial coefficients. The proof requires a new moderate-growth estimate for de Crisenoy's fully twisted zeta functions, proved in Section 3 as Proposition 6. Several worked examples show the formula in action, including an example where the special values are transcendental.","tokens_in":29285,"tokens_out":37964,"duration_ms":349008,"significance":"The result is a substantial extension of the authors' Part I, where only linear and power-sum denominators were treated. The explicit special-value formula (6.2) is new and is expressed in a usable form, and the companion moderate-growth result (Proposition 6) is a useful complement to de Crisenoy's theorem. The transcendental example (Example 35) is particularly valuable because it shows that the arithmetic nature of the values changes genuinely with the shape of the polynomial denominator. The paper is carefully written and the main derivation is internally consistent; no circularity or fitted parameters are present.","major_comments":[],"minor_comments":[{"comment":"The sentence 'Since at each stage of his recurrence, de Crisenoy proceeds only by integration by parts, it is clear that his proof implies...' compresses the uniformity argument in a way that is load-bearing for Proposition 6. Please spell out explicitly how the induction constants remain uniform in the functions f_k (and not merely in the data Q, R_t), and why the coefficients produced by the recurrence have at most polynomial growth in s. As written, this is the only place where an unstated uniformity is required.","section":"Section 3.2, proof of Proposition 15"},{"comment":"The displayed limit evaluates Γ(eg(s_T;-ℓ))/Γ(s_T), but the term being analyzed contains the product Γ(eg(s_T;-ℓ))Γ(g(s_T;-ℓ))/Γ(s_T). Please add an explicit sentence explaining that Γ(g(-N_T;-ℓ)) = (|ℓ|+i-N_T-1)! and that the product limit is what produces the second sum in (6.2). Without this clarification, the transition from the limit to the final formula is not fully visible.","section":"Section 7.4, proof of Theorem 24"},{"comment":"When applying de Crisenoy's Proposition 3 to the fully twisted family P(Q) = (P_1,...,P_{T-1},Q_0,...,Q_d), the hypotheses require the product of all entries to tend to infinity. The authors assume (2.4) instead. Please add a sentence explaining that (2.4) implies the needed growth condition, e.g. by noting that each positive polynomial has positive infimum on [1,∞)^{n-1}.","section":"Section 6"},{"comment":"There are a few small typographical slips: a double comma in the definition of P_{n1,n2,n3}^ε in Section 3.3, and a missing closing parenthesis in the first line of Example 35. These should be corrected.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The central argument is sound and the paper is a strong continuation of the authors' earlier work. The only points that warrant attention are presentation-level: the uniformity step in Proposition 15 and the product-limit step in the proof of Theorem 24 should be made explicit. I found no mathematical error in the main reduction or in the examples."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know that this paper is a genuine extension, not a repackaging: it takes the partially twisted multiple zeta case with k=n-1 and handles arbitrary polynomial denominators, with T not necessarily equal to n. Previous work covered only linear forms and power sums. The main formula (Theorem 24) expresses the non-positive integer values via de Crisenoy's fully twisted values, twisted Bernoulli numbers, and multinomial coefficients. I read the reduction carefully and the Mellin-Barnes machinery works. The new ingredient is Proposition 6, a moderate-growth estimate for de Crisenoy's zeta function; without it the repeated contour shifts don't justify the holomorphy of the remainders. The proof adapts de Crisenoy's integration-by-parts induction, and the uniform B(1,M) argument seems to supply the needed decay.\n\nThe paper steps well: it does d=1, d=2, then the general case, and the examples are useful. The example with n=T=2 giving values as rational polynomials in pi is a clean transcendence demonstration once the reduction is accepted.\n\nSoft spots are proportionate. The proof of Proposition 15 is compressed at the point where it inherits de Crisenoy's induction and asserts uniform constants for iterated primitives. I had to supply details; it looks fixable, but a referee should ask for a few sentences more there. The residue computations in Sections 4-7 are long, and I didn't verify every Bernoulli/gamma identity line by line. The recursive structure and the endpoint consistency are convincing, and the special value limit computations in 7.4 line up with the d=1 and d=2 cases. Another limitation: the formulas are reductions rather than closed forms, because fully twisted values at positive integer arguments can appear. The paper says this candidly; it's a limitation on explicitness, not a correctness problem.\n\nCitation pattern is fine: de Crisenoy is the external input, their previous paper is motivation, and the related literature is cited. I found no fitted parameters, no circularity, and no claim that the formulas are elementary when they are not.\n\nIf I were the editor, I would send this to a specialist referee. It is a serious piece of work. My own verdict is accept. Not every reader needs this, but anyone working on twisted multiple zeta values or Mellin-Barnes reductions will use it.","headline":"A solid, genuinely new reduction of the partially twisted multiple zeta case with general polynomial denominators to de Crisenoy's fully twisted theorem; I found no fatal flaw.","tokens_in":29815,"tokens_out":2657,"would_cite":true,"duration_ms":27439,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","11M35","11M41"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an explicit formula for the values at non-positive integer points of partially twisted multiple zeta-functions with general polynomial denominators, expressing them through de Crisenoy's fully twisted zeta values at…","keywords":["partially twisted multiple zeta-functions","polynomial denominators","Mellin-Barnes integral formula","values at non-positive integers","Lerch zeta-function","Bernoulli numbers","meromorphic continuation","transcendence"],"falsifier":"Set n=T=1 and P(X)=1+X in the formula of Section 8.2; since zeta_1(s;P;1) equals zeta(s)-1 for this P, comparing the right-hand side at s=-N with the known value zeta(-N)=-B_{N+1}/(N+1) at any N>=0 would immediately detect an error in Theorem 24.","tokens_in":28928,"feed_emoji":"🔢","tokens_out":9699,"duration_ms":91417,"temperature":0.7,"pith_summary":"Partially twisted multiple zeta-functions have twisting characters in all summation variables except the last, and denominators that are products of polynomial factors. This paper establishes that for general polynomial denominators satisfying the HDF positivity and decay condition, every non-positive integer tuple is a regular point, and the value there is given by an explicit finite formula. The formula expresses the value in terms of de Crisenoy's fully twisted zeta values at integers, twisted Bernoulli numbers, and multinomial coefficients. A consequence visible in the paper's examples is that these values can be transcendental, such as non-constant rational polynomials in pi. The proof works by a Mellin-Barnes separation of the last summation variable and iterative contour shifts, resting on a new moderate-growth estimate for the fully twisted zeta-function.","feed_headline":"Explicit formula found for partially twisted zeta values","feed_subtitle":"General polynomial denominators reduce to fully twisted zeta values plus Bernoulli numbers.","key_machinery":"The load-bearing mechanism is the multiple Mellin-Barnes integral formula (Proposition 8), applied to the inner sum over m_n so that the argument of the resulting zeta factor depends on new integration variables z. Iterative left shifts of the contour in the variables z_j produce the auxiliary integrals $I^{{(r)}}$ and $J^{{(r)}}$ (Definitions 28-29), whose residue calculus reduces everything to values of de Crisenoy's fully twisted zeta-function at integers. All shifts are justified by Proposition 6, the new proof that the fully twisted zeta-function is of moderate growth; the uniform estimate behind it is Proposition 15.","core_discovery":"The central claim is Theorem 24: under the HDF and growth hypotheses on the polynomials P_1,...,P_T, for every N=(N_1,...,N_T) with non-negative integer entries the special value zeta_n(-N;P;mu_{n-1}) equals the right-hand side of formula (6.2), a finite sum of products of multinomial coefficients, twisted Bernoulli numbers, and values of the fully twisted zeta-function $zeta^{{dC}}$_{n-1} at integer points. Corollary 23 states that zeta_n(s;P;mu_{n-1}) continues meromorphically to all of C^T with singularities contained in explicit hyperplanes, and none of the non-positive integer points is singular. The same Mellin-Barnes mechanism yields the intermediate d=1 and d=2 cases (Theorems 17 and 18) and recovers the expected linear-case formulas.","pith_inferences":["A direct test of the formula in the one-variable case P(X)=1+X, where zeta_1(s;P;1) equals zeta(s)-1, would verify the residue calculus in a clean setting without needing any numerical analytic continuation.","If Proposition 6's moderate-growth bound can be made explicit with constants, the Mellin-Barnes representation (7.1) could yield uniform estimates for these zeta-functions in vertical strips, which would be useful for zero or value-distribution questions.","The appearance of fully twisted values at rational arguments such as (a_1 k_2+1)/a_2 in the d=2 case suggests a broader pattern: partial twisting may force special values to be governed by Lerch-type values at rational points, linking to polylogarithms at roots of unity.","The same separation-of-variables trick should adapt to partial twists with more than one untwisted factor, provided the moderate-growth estimate holds for the corresponding fully twisted function with those extra variables."],"forward_implications":["Every non-positive integer point is a regular point of zeta_n(s;P;mu_{n-1}), so no limiting procedure is needed at those points.","The special values are completely determined by fully twisted zeta values at integers, twisted Bernoulli numbers, and multinomial coefficients, so no new transcendental constants enter beyond those already present in the fully twisted case.","When the leading coefficient of P_T is constant, the special values lie in the field generated over Q by the roots of unity mu_j and the coefficients of the polynomials, meaning no new transcendence appears.","In the example P_1=1, P_2=1+X_2+X_1^q X_2^2 with mu=-1 and q even, the values are non-constant rational polynomials in pi and are therefore transcendental.","The authors state that the same method should extend inductively to cases with fewer twists (k <= n-2), in the style of their earlier linear and power-sum treatment."],"supporting_citations":[{"why":"Supplies the fully twisted case: entire continuation, the explicit special-value formula (2.3), and the method extended here to prove moderate growth.","marker":"[4]"},{"why":"Supplies the H0S condition and Lemma 1 used to prove the absolute convergence domain (Proposition 9) of the series (1.1).","marker":"[6]"},{"why":"The source of the multiple Mellin-Barnes formula quoted as Proposition 8, the core tool used to separate m_n from the other summation variables.","marker":"[16]"},{"why":"Provides the single Mellin-Barnes formula used in Section 4 for the d=1 case (Proposition 7).","marker":"[23]"}],"fun_headline_variants":["Explicit formulas for partially twisted zeta values","Reduction to fully twisted gives explicit zeta values","Polynomial denominators: explicit values of twisted zetas","Transcendental values at non-positive integers for twisted zetas","Explicit values for multi-variable twisted zeta functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the new moderate-growth estimate for de Crisenoy's fully twisted zeta-function; if that estimate failed for some polynomial denominator satisfying the paper's HDF positivity and decay condition, the contour shifts that produce the explicit formula would not be justified.","fun_headline_variants_meta":{"raw":{"variants":["Explicit formulas for partially twisted zeta values","Reduction to fully twisted gives explicit zeta values","Polynomial denominators: explicit values of twisted zetas","Transcendental values at non-positive integers for twisted zetas","Explicit values for multi-variable twisted zeta functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1308,"prompt_tokens":840,"completion_tokens":468,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":389}},"tokens_in":456,"tokens_out":468,"duration_ms":5380,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:22:57.385312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set n=T=1 and P(X)=1+X in the formula of Section 8.2; since zeta_1(s;P;1) equals zeta(s)-1 for this P, comparing the right-hand side at s=-N with the known value zeta(-N)=-B_{N+1}/(N+1) at any N>=0 would immediately detect an error in Theorem 24.","supporting_citations":[{"cited_title":"de Crisenoy, Values atT-tuples of negative integers of twisted multi- variable zeta series associated to polynomials of several variables, Com- positio Math.142(2006), no","cited_arxiv_id":null,"evidence_quote":"Supplies the fully twisted case: entire continuation, the explicit special-value formula (2.3), and the method extended here to prove moderate growth."},{"cited_title":"Essouabri, Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et application en th´ eorie analytique des nombres, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the H0S condition and Lemma 1 used to prove the absolute convergence domain (Proposition 9) of the series (1.1)."},{"cited_title":"Mellin, Eine Formel f¨ ur den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Soc","cited_arxiv_id":null,"evidence_quote":"The source of the multiple Mellin-Barnes formula quoted as Proposition 8, the core tool used to separate m_n from the other summation variables."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the single Mellin-Barnes formula used in Section 4 for the d=1 case (Proposition 7)."}],"review_version":2}