{"id":"4a5dd334-d0c3-46d8-b3ee-29a50718ad30","arxiv_id":"2507.23619","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generating-function framework solves convolution-like recurrences and constructs b-sequences whose alpha-sequences are partial sums of zeta values, pi, e, and many OEIS sequences.","lead":"This paper develops a general method for solving convolution-like recurrences that relate an unknown sequence a to a known sequence b, and it shows that by choosing b one can generate partial sums of familiar constants such as zeta values, pi, and e. The work's interest for a generalist reader lies in its clean generating-function framework and in its claim of a new equivalent condition for the Riemann hypothesis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3's RH criterion is unsupported: the equivalence between ordinary convergence of ∑ j b̂_j(a) and non-vanishing of ζ(a) in the strip is exactly the unproved step, and Proposition 3's derivation is omitted.","rationale":"I read the paper as primarily a generating-function framework for the recurrence (1), with the RH criterion in Corollary 3 as the attention-grabbing application. Theorem 1, Proposition 1, and the derivations of α-sequences in Propositions 2 and 4 are internally coherent; the OEIS checks and concrete examples provide genuine supporting evidence for the framework. The reader's weakest assumption is exactly where I also find the soft spot. My concern is more specific than an omitted proof: Proposition 3's hypotheses are analytic-continuation conditions, while Corollary 3's hypothesis is ordinary convergence of a coefficient series. The displayed identity equating the two is the central unproved step; without a Tauberian argument or a coefficient-decay bound, Corollary 3 is not established. I also note that the proof of Proposition 3 explicitly defers the derivation, and Note 3's decay comment is only a sufficient condition. Because the framework itself is not implicated, I do not call for rejection; the conditional verdict already reflects that the advertised RH claim needs full proof or explicit conjecture status.","tokens_in":23687,"tokens_out":19005,"duration_ms":196654,"concrete_test":"Independently re-derive Proposition 3 from the recurrence defining b̂_n, without invoking Theorem 2's limit interchange, and identify where ordinary convergence of ∑_{n≥1} n b̂_n(a) is proved. In parallel, run a numerical check: fix a=3/4+14i, compute b̂_1,...,b̂_N via the Proposition 3 recurrence for N=10^5, form S_N=∑_{n=1}^N n b̂_n(a), and compare with 1−1/((2−2^{2−a})ζ(a)). If |S_N−limit| does not go to 0 while ζ(a) is finite and nonzero, the 'if' direction of Corollary 3 fails; if it converges, the equivalence still needs the missing Tauberian estimate to be promoted from one sample to the whole strip. A complementary check is to compute the radius of convergence of B̂(s)=s+(1−s)/D_a(s) at this a: radius >1 would make ordinary convergence plausible, while radius ≤1 would show that the hypotheses of Proposition 3 do not by themselves guarantee it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing claim is Corollary 3: RH iff ∑_{j≥1} j b̂_j(a) converges in 1/2<Re a<1. This is not a corollary of the stated Proposition 3. Proposition 3 assumes that f(s,a) is analytic in |s|<1 and that lim_{s→1-} f(s,a)=0; under those hypotheses it concludes ζ(a)=1/((2-2^{2-a})(1-∑_{j≥1} j b̂_j(a))). The hypotheses are statements about analytic continuation of a generating function; they do not imply that the ordinary coefficient series ∑ j b̂_j(a) converges. Passing from the value of the generating function at s=1 to the ordinary sum is a Tauberian step, and no such argument is supplied. Consequently Corollary 3 either requires a new convergence theorem for the b̂_j coefficients or is a restatement of the identity in Proposition 3, not an independent criterion. The proof of Proposition 3 itself says 'After long and careful derivation (we omit details)', so the crucial derivation is unavailable in the manuscript. This concern does not affect Theorem 1, the α-sequence framework, or the OEIS examples, but it controls the paper's most advertised statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the convolution-like recurrence (1), a_n = sum_{j=0}^{n+m} b_{n+m-j} a_j, with known b and b_0 ≠ 0. It introduces m auxiliary sequences α_0,…,α_{m-1} so that a_n is a linear combination of a_0,…,a_{m-1} (Proposition 1), derives generating-function identities for these sequences (Theorem 1), gives conditions for the existence of limits and explicit limit formulas (Theorem 2), and provides a linear system for recovering a_0,…,a_{m-1} from lim a_n (Theorem 3 and Corollary 2). Propositions 2–4 construct sequences b whose α-sequences are partial sums of the Riemann zeta function, π/4, and e, and express the corresponding constants as ∑ j b_j. The most advertised consequence is Corollary 3, which states that the Riemann hypothesis is true if and only if ∑ j b̂_j(a) converges in 1/2 < Re a < 1 for the coefficients b̂_j of Proposition 3.","tokens_in":23884,"tokens_out":3635,"duration_ms":42000,"significance":"If the main framework is correct, Theorem 1 and Theorem 2 provide a clean, elementary generating-function method for solving a natural family of recurrences, and the OEIS examples in Section 4 give the paper a useful computational character. The direct coefficient manipulations in the proofs of Proposition 1 and Theorem 1 are plausible and accessible, and the limit formulas in Theorem 2 have the virtue of being explicit and checkable. However, the paper's most striking claim, the Riemann-hypothesis criterion in Corollary 3, is not supported by the arguments actually supplied: the step from a boundary value of a generating function to the ordinary convergence of the coefficient series is a nontrivial Tauberian statement, and the derivation of Proposition 3 is explicitly omitted. The paper therefore mixes a sound-looking core with an advertised result that is currently unproved.","major_comments":[{"comment":"Corollary 3 is not a consequence of the stated Proposition 3. Proposition 3 assumes that f(s,a) in (26) is analytic in |s|<1 and that lim_{s→1-} f(s,a)=0; under those hypotheses it derives the identity ζ(a)=1/((2-2^{2-a})(1-∑_{j≥1} j b̂_j(a))). The identity concerns the value at s=1 of a generating function, while the corollary concerns the ordinary convergence of the coefficient series ∑ j b̂_j(a). Passing from the former to the latter is a Tauberian step requiring coefficient estimates or a convergence theorem, and no such argument appears. Thus the equivalence 'RH iff ∑ j b̂_j(a) converges in 1/2<Re a<1' is currently an unsupported assertion, not a proved criterion.","section":"§2, Corollary 3"},{"comment":"The proof of Proposition 3 consists of the sentence 'After long and careful derivation (we omit details)' followed by the claimed formal series B̂(s). Since B̂(s) defines the coefficients b̂_j on which Corollary 3 depends, this omission is load-bearing. The full derivation of B̂(s), including the identities for its coefficients, must be supplied in the manuscript or replaced by a complete reference with proof.","section":"§3, Proof of Proposition 3"},{"comment":"Theorem 3's uniqueness claim rests on the determinant formula (30), which is cited from [7, Lem. 4.2] rather than proved, and Corollary 2's proof is deferred to [7, Proof of Thm. 3.3]. Because the invertibility of the system (17) is essential for the claimed inversion of initial values via lim a_n, the paper should either prove these determinant and minor identities or state the cited results in enough detail that the proof is self-contained. As written, a substantial part of the advertised inversion machinery lives in a previous paper.","section":"§2, Theorem 3 and Corollary 2"}],"minor_comments":[{"comment":"Note 2 refers to 'Corollary (1)' but the intended cross-reference is Corollary 1; the notation should be corrected.","section":"§2, Note 2 and Corollary 1"},{"comment":"The derivation of (9) from (28) equates coefficients of formal power series; the argument would be clearer if it explicitly stated that (28) is an identity of formal power series before invoking the Maclaurin expansion, since some of the displayed expressions are not known a priori to converge.","section":"§3, proof of Theorem 1"},{"comment":"In equation (25) the condition 'ℜα >1' appears at the end; this should be 'ℜa>1' as in the rest of the proposition.","section":"§2, Proposition 2"},{"comment":"The figures in Section 2 and Section 4 are visually striking, but the axes are unlabeled and the plotted quantities are identified only in the captions; adding explicit axis labels and a short description of the plotted range would allow readers to verify the 'picturesque' patterns independently.","section":"§4, Examples and figures"},{"comment":"The list of OEIS matches in §4.3 gives b sequences and the resulting a sequences without derivation or a proof that a = α_0 in each case; a sample derivation for at least two entries, plus a note on how the remaining entries are verified, would make the section reproducible.","section":"§4.3, list of examples"},{"comment":"The condition 'b_0 + b_1 + … = 1' is used in (15) as if it were an ordinary convergent series, while elsewhere the paper works with formal power series; the analytic convergence assumptions should be stated explicitly in Theorem 2.","section":"§2, Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The paper has a useful elementary core (Theorem 1, Theorem 2, and the OEIS examples) that can likely be made publishable. The main risk is the Riemann-hypothesis corollary: it is advertised prominently but depends on an unproved Tauberian equivalence and on an omitted derivation in Proposition 3. If the authors can either provide a genuine convergence theorem for ∑ j b̂_j(a) or explicitly downgrade Corollary 3 to a formal restatement, the paper would be much stronger. I would also suggest that the editors weigh whether the determinant and corollary proofs borrowed from [7] should be reproduced for self-containedness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the α-sequence generating-function framework is real and the extension from probability distributions to arbitrary b is useful, but Corollary 3's Riemann hypothesis criterion is not supported by the paper's own arguments. That advertised claim should be withdrawn until the missing analysis is supplied.\n\nThe core content is Theorem 1: the recurrence a_n = Σ b_{n+m−j} a_j is solved in closed form via (5)–(7). These are standard coefficient manipulations and they look correct. Generalizing the earlier m=2 probabilistic recurrences to general m and b with b0 ≠ 0 is natural and clean. The examples—Lucas, Catalan, Bell, Gould, and the partial-sum identities for ζ, π, e—are numerous, and the OEIS matches are a reasonable sanity check. For anyone working with convolution recurrences, this is a usable toolkit.\n\nThe soft spots sit in the Riemann hypothesis material. Proposition 3 is supposed to derive the generating function for the Hasse coefficients, but the proof says “After long and careful derivation (we omit details).” That means the central formula is not actually derived in the paper. Corollary 3 then claims RH is equivalent to convergence of Σ j b̂_j(a) in the strip. The stress-test note is right: the hypotheses of Proposition 3 (analyticity of f(s,a) in |s|<1 and lim f(s,a)=0 as s→1−) do not imply the ordinary coefficient series converges at s=1. That passage is a Tauberian step and no argument is supplied. So the RH criterion is an unproved equivalence, and it carries the full difficulty of RH. This does not undermine Theorem 1 or the examples, but it means the paper's most consequential statement should not be relied on.\n\nThere is also a self-containment issue: Theorem 3's determinant and Corollary 2's minors are imported from the authors' earlier paper [7] rather than proved here. That is acceptable if the citation holds up, but it makes the paper less independent. The plots are illustrative; missing error bars are a minor point for a math paper.\n\nWho this is for: researchers in recurrences and generating functions, and readers who enjoy seeing famous sequences encoded as coefficients. The framework deserves a serious referee. My recommendation: send it to review, but require the authors either to give a complete proof of Proposition 3 and the convergence equivalence, or to delete Corollary 3 and the RH claims. The rest can stand alone.","headline":"The generating-function framework is a solid, useful generalization, but the advertised Riemann hypothesis criterion is unproved and should be withdrawn or fully proved.","tokens_in":36,"tokens_out":3400,"would_cite":true,"duration_ms":93380,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B37","11M06","30B10","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Riemann hypothesis is restated as convergence of a coefficient series.","keywords":["convolution-like recurrence","linear recurrence","initial values","Maclaurin series","power series","partial sums","Riemann hypothesis","generating functions"],"falsifier":"Compute $\\hat b_n(a)$ from Proposition 3 for several $a$ with $1/2<\\Re a<1$ and numerically check convergence of $\\sum_{j=1}^\\infty j\\hat b_j$; a single point where convergence disagrees with the zero-free status of $(2-2^{2-a})\\zeta(a)$ would disprove Corollary 3. Concretely, finding a zero of $(2-2^{2-a})\\zeta(a)$ in the strip at which the coefficient series converges would already settle it.","tokens_in":23440,"feed_emoji":"🔢","tokens_out":12053,"duration_ms":111472,"temperature":0.7,"pith_summary":"The paper establishes a closed-form generating-function method for the convolution-like recurrence $a_n=\\sum_{j=0}^{n+m} b_{n+m-j}a_j$ with a known complex sequence $b$, $b_0\\ne 0$. The key consequence of Theorem 1 is the identity $G_{\\alpha_k}(s)(B(s)-s^m)=\\sum_{n=k}^{m-1} b_{n-k}s^n$, so each auxiliary sequence $\\alpha_k$---and therefore every solution $a$---is read off from $B(s)$ by extracting coefficients. From this the authors derive when the limit $\\lim_{n\\to\\infty} a_n$ exists, express the initial values $a_0,\\dots,a_{m-1}$ back through that limit, and show that a suitable choice of $b$ makes $a$ many known sequences, including partial sums of $\\zeta(a)$, $\\pi$, and $e$. A final corollary claims that the Riemann hypothesis holds if and only if the coefficient series $\\hat b_1+2\\hat b_2+\\cdots$ converges in the strip $1/2<\\Re a<1$.","feed_headline":"Riemann hypothesis becomes a convergence question","feed_subtitle":"A generating-function identity solves the whole recurrence family and ties ζ(a) to a coefficient series.","key_machinery":"The load-bearing object is the auxiliary sequence family $\\alpha_0,\\dots,\\alpha_{m-1}$, each defined by the same recurrence (3) with unit initial vectors, together with their generating functions $G_{\\alpha_k}(s)=\\sum_{n=0}^\\infty \\alpha_k(n)s^n$. The identity that carries the argument is $G_{\\alpha_k}(s)(B(s)-s^m)=\\sum_{n=k}^{m-1} b_{n-k}s^n$, which makes every $\\alpha_k$ the coefficient sequence of an explicitly known power-series quotient once $B(s)$ is known. This single identity yields the representation of arbitrary solutions, the Abelian limit theorem, the linear system in Theorem 3, and the selection of $b$ that produces famous partial sums.","core_discovery":"The central discovery is that the recurrence (1) is coefficient extraction from a quotient, not a genuinely infinite problem. Writing $A(s)=\\sum a_n s^n$ and $B(s)=\\sum b_n s^n$, Theorem 1 proves $A(s)(B(s)-s^m)=\\sum_{k=0}^{m-1}a_k\\sum_{n=k}^{m-1}b_{n-k}s^n$, and for each basis sequence $\\alpha_k$ the same manipulation gives $G_{\\alpha_k}(s)(B(s)-s^m)=\\sum_{n=k}^{m-1}b_{n-k}s^n$. Proposition 1 shows $a_n=\\sum_{k=0}^{m-1}\\alpha_k(n)a_k$, so the whole solution space is $m$-dimensional and explicitly generated. Theorem 2 gives a closed form for $\\lim_{n\\to\\infty}\\alpha_k(n)$ under the Abelian condition that the Maclaurin series of $(1-s)G_{\\alpha_k}(s)$ converge at $s=1$, and when $\\sum b_j=1$ and $m\\ne\\sum j b_j$ the limit equals $(\\sum_{j=k}^{m-1}b_{j-k})/(m-\\sum_{j=1}^\\infty j b_j)$. Theorem 3 and Corollary 2 solve for the initial values from $\\lim a_n$ via a linear system whose determinant is a Vandermonde-type product over the roots of $B(s)-s^m$. The applications choose $b$ so that $\\alpha_0(n)$ is a prescribed partial sum: harmonic sums of $\\zeta(a)$, Möbius-weighted sums, Hasse-type coefficients that evaluate $\\zeta(a)$ in the whole plane, Leibniz sums for $\\pi/4$, and exponential sums for $e$. From the Hasse-based choice the paper derives Corollary 3: the Riemann hypothesis is true exactly when $\\hat b_1+2\\hat b_2+\\cdots$ converges in $1/2<\\Re a<1$.","pith_inferences":["Inference: the same convolution inversion should apply to recurrences with a different shift or with an added forcing term, because the generating-function manipulation only uses the shift structure of the convolution.","Inference: for $m=1$, the construction gives a dictionary between sequences $b$ and sequences $a$: any sequence whose generating function $A(s)$ satisfies $A(s)(B(s)-s)=a_0 b_0$ is generatable, so the reachable sequences are exactly coefficients of reciprocals of power series of the form $B(s)-s$.","Inference: the equivalence in Corollary 3 can be probed numerically before any proof: if a point $a$ in the strip emerges where $\\sum j\\hat b_j$ diverges while $(2-2^{2-a})\\zeta(a)$ is known to be nonvanishing, the corollary would be false; no such computation would, on its own, prove RH.","Inference: the paper's plots of $(\\alpha_0(n),\\alpha_0(n+1))$ track the radius of convergence of $G_{\\alpha_0}$; the growth of successive coefficients in those plots should follow the reciprocal of the nearest singularity of $B(s)-s^m$, which Corollary 1 predicts."],"forward_implications":["Any solution of the recurrence can be computed by extracting coefficients from $B(s)$, bypassing step-by-step iteration of the recurrence.","When $\\sum_j b_j=1$ and $m\\ne\\sum_j j b_j$, the limit $\\lim_{n\\to\\infty} a_n$ is a finite rational expression in the first $m$ initial values and the known coefficients $b_j$.","The initial values $a_0,\\ldots,a_{m-1}$ are recoverable from the single number $\\lim_{n\\to\\infty}a_n$ by solving the linear system of Theorem 3, with explicit closed forms in Corollary 2.","Choosing $b$ appropriately generates many named sequences as $a$—Lucas, Bell, Catalan, Motzkin, Ramanujan tau, and the partial sums of $\\zeta(a)$, $\\pi$, and $e$.","If Corollary 3 holds, the Riemann hypothesis is equivalent to the convergence of $\\hat b_1+2\\hat b_2+\\cdots$ on the strip $1/2<\\Re a<1$."],"supporting_citations":[{"why":"Supplies the determinant lemma (its Lem. 4.2) that proves the linear system in Theorem 3 has a unique solution and gives determinant (30).","marker":"[7]"},{"why":"Provides the Hasse summation formula for $\\zeta(a)$, which Proposition 3 and Corollary 3 rely on to evaluate zeta from coefficients.","marker":"[10]"},{"why":"Introduced recurrences of this type with $m=2$ in a probabilistic risk model, the setting this paper generalizes to arbitrary $b$.","marker":"[3]"},{"why":"Proved the relation between recurrences of type (3) and probability generating functions, motivating the generating-function machinery.","marker":"[8]"},{"why":"Gives the Maclaurin-series and radius-of-convergence facts used in Corollary 1.","marker":"[1]"},{"why":"Gives the limit law for maxima of discrete partial sums used in the probabilistic interpretation of Example 6.","marker":"[9]"}],"fun_headline_variants":["Convolution recurrences solved via generating functions","Partial sums of zeta from a single coefficient series","Riemann hypothesis becomes a convergence test","Recurrence family reduces to m initial values"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that convergence of the coefficient series $\\hat b_1+2\\hat b_2+\\cdots$ in the strip is equivalent to the analytic function $(2-2^{2-a})\\zeta(a)$ being zero-free there; the paper asserts this equivalence after Proposition 3 but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["Convolution recurrences solved via generating functions","Partial sums of zeta from a single coefficient series","Riemann hypothesis becomes a convergence test","Recurrence family reduces to m initial values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000775,"raw_usage":{"total_tokens":3541,"prompt_tokens":1170,"completion_tokens":2371,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":786,"completion_tokens_details":{"reasoning_tokens":2314}},"tokens_in":786,"tokens_out":2371,"duration_ms":19325,"temperature":1.0,"reasoning_tokens":2314,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:30:46.661626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\hat b_n(a)$ from Proposition 3 for several $a$ with $1/2<\\Re a<1$ and numerically check convergence of $\\sum_{j=1}^\\infty j\\hat b_j$; a single point where convergence disagrees with the zero-free status of $(2-2^{2-a})\\zeta(a)$ would disprove Corollary 3. Concretely, finding a zero of $(2-2^{2-a})\\zeta(a)$ in the strip at which the coefficient series converges would already settle it.","supporting_citations":[{"cited_title":"Exact expression of ultimate time survival probability in homo- geneous discrete-time risk model","cited_arxiv_id":null,"evidence_quote":"Supplies the determinant lemma (its Lem. 4.2) that proves the linear system in Theorem 3 has a unique solution and gives determinant (30)."},{"cited_title":"Ein summierungsverfahren f¨ ur die riemannscheζ-reihe","cited_arxiv_id":null,"evidence_quote":"Provides the Hasse summation formula for $\\zeta(a)$, which Proposition 3 and Corollary 3 rely on to evaluate zeta from coefficients."},{"cited_title":"Bi-seasonal discrete time risk model","cited_arxiv_id":null,"evidence_quote":"Introduced recurrences of this type with $m=2$ in a probabilistic risk model, the setting this paper generalizes to arbitrary $b$."},{"cited_title":"On 2 ×2 determinants originating from sur- vival probabilities in homogeneous discrete time risk model","cited_arxiv_id":null,"evidence_quote":"Proved the relation between recurrences of type (3) and probability generating functions, motivating the generating-function machinery."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Maclaurin-series and radius-of-convergence facts used in Corollary 1."},{"cited_title":"The limit law of maximum of discrete partial- sums distribution","cited_arxiv_id":null,"evidence_quote":"Gives the limit law for maxima of discrete partial sums used in the probabilistic interpretation of Example 6."}],"review_version":1}