{"id":"2286c351-379e-4502-a560-bbdfe8fd492c","arxiv_id":"2507.07253","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Riemann sequences are defined to mimic the zeta-zero asymptotic; the zeta zeros form one, infinitely many others are constructed via crystalline measures, and the paper asks whether the zeta sequence is the unique real one.","lead":"This paper introduces a class of sequences, called Riemann sequences, that share the same asymptotic zero-counting law as the imaginary parts of the nontrivial zeros of the Riemann zeta function. It constructs infinitely many such sequences, gives a concrete non-real example, and conjectures that the zeta zeros form the only real Riemann sequence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The existence theorem for Riemann sequences is analytically sound, but the concrete complex example in Section 6 relies on a non-rigorous x-ray and an unverified zero-free half-plane (σ0=10.564...); without rigorous error control or explicit tail bounds, the claim that non-real Riemann sequences…","rationale":"We read the paper as making two principal claims: (1) Theorem 14 constructs infinitely many Riemann sequences via crystalline measures and a perturbation of ζ; (2) Section 6 exhibits a concrete complex Riemann sequence. We checked the proof of Theorem 14 in detail. The dimension argument in Propositions 8–9, the analytic continuation via Theorem 5, the zero-free regions in Proposition 12, and the product representation in Proposition 13 all hold together; in particular, the zero-free region in Prop 12, together with evenness and conjugation symmetry, indeed forces Re α>3/2 and |Im α|<3/2 for zeros with Re α>0. The step from the product representation and ζ_N(s)≃1 to the asymptotic expansion (16) is a faithful analogue of Section 2; no hidden dependence on the residue at s=1 appears after differentiation. Thus the central existence theorem is sound. The single load-bearing weakness is the Section 6 demonstration of a non-real Riemann sequence. The zero-free half-plane σ0=10.564... is presented without a rigorous tail bound; the x-ray and zero table are not certified. Because condition (a) and (d) of Definition 4 require control of all zeros with Re α>0, a missed zero with small ordinate would break the Riemann-sequence property. This is precisely the reader's weakest assumption. Our proposed check—a verified argument-principle count and an interval-arithmetic tail bound—would settle whether the claimed complex example is genuine. If the check fails, the paper should be accepted only after softening the Section 6 claim to numerical evidence; if it passes, the conditional verdict can be upgraded. We set verdict_should_be to UNCHANGED because our assessment matches the reader's conditional recommendation.","tokens_in":17891,"tokens_out":33736,"duration_ms":347615,"concrete_test":"Run a certified computation for the Section 6 example: (1) using interval arithmetic (e.g., Arb), evaluate an explicit upper bound for ∑_{n≥2}|c_n|λ_n^{-σ} over σ≥σ0 and check it is <1 at σ0=10.564029176912431172; (2) compute the argument-principle integral along ∂R with rigorous error bounds and compare the resulting zero count with 31 (minus the pole at s=1); (3) verify each found zero satisfies Re α>1 and |Im α|<Re α. If all three pass, the concrete complex Riemann sequence is established; if any step fails, the claim should be downgraded to numerical evidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main structural result, Theorem 14, is derived from the product representation (26) and the zero-free regions of Proposition 12; those arguments appear internally consistent. The load-bearing weakness is the Section 6 numerical construction of ζ_M. To conclude that its zeros form a complex Riemann sequence, one needs (i) a rigorous zero-free half-plane σ≥σ0, (ii) a certified count of zeros in the rectangle R=(-21,22)×(-10,80) via the argument principle, and (iii) a proof that all zeros with Re α>0 satisfy Re α>1 and |Im α|<Re α. The paper supplies only the heuristic x-ray (Figure 1), a bare number σ0=10.564029176912431172 with no tail bound or interval arithmetic, and a table of 31 zeros in R without error control. In particular, the phrase 'where the dots represents all the other terms in ζ_M(s) taken with coefficients in absolute value' is not a bound; it is an assertion that the infinite tail has been evaluated. A missed zero with small ordinate would violate condition (a) or (d) of Definition 4, and would invalidate the Riemann-sequence claim. Since the asymptotic expansion (16) itself is a formal consequence of the product representation and ζ_M(s)≃1, the only missing link is this numerical verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of a Riemann sequence: a sequence of complex numbers satisfying the same asymptotic expansion as the nontrivial zeros of the Riemann zeta function, namely sum_k 2z/(z^2+alpha_k^2) ~ (1/2)log(z/2pi) + sum_n a_n/z^n with the coefficients a_n given by Euler and Bernoulli numbers, together with structural axioms on ordering, symmetry, imaginary parts, and a sector condition. The authors prove that the zeta zero parameters tau_n form a Riemann sequence (Theorem 3), construct infinitely many Riemann sequences via crystalline measures and finite Fourier analysis (Theorem 14), present an explicit candidate complex Riemann sequence attached to a combination of Hurwitz zeta functions (Section 6), and derive common analytic properties of the associated Xi, zeta, and secondary zeta functions (Sections 7 and 8). The advertised program is to characterize the zeta zeros as the unique real Riemann sequence.","tokens_in":18139,"tokens_out":9783,"duration_ms":109149,"significance":"If the existence of complex Riemann sequences were rigorously established, this would give a new intrinsic characterization of the zeros of the Riemann zeta function, independent of the Riemann hypothesis, and would connect this problem to the theory of crystalline measures. The analytic core of the paper is sound: Theorem 1 and the derivation of the asymptotic relation from the classical product formula, the product-representation argument behind Theorem 14, and the structural study of Riemann sequences in Sections 7 and 8 are coherent and clearly presented. The paper is honest that the zeta zeros themselves are used as a source of properties rather than as an assumed conclusion, and it gives explicit formulas for residues and special values of the secondary zeta function Z_alpha. The main weakness is Section 6: the sole evidence for the existence of a complex Riemann sequence is a numerical computation whose completeness and error control are not certified, and the paper's central advertised claim therefore currently exceeds what is proved.","major_comments":[{"comment":"The zero-free half-plane sigma >= sigma_0 for the function zeta_M is not proved. The displayed equation 'This happens for sigma_0 = 10.564029176912431172' is preceded by an expression ending with '... where the dots represents all the other terms in zeta_M(s) taken with coefficients in absolute value'. An ellipsis with absolute values is not a tail bound. A rigorous argument requires an explicit dominating convergent series, interval arithmetic, or an effective bound on the remaining terms of the Dirichlet expansion. This matters because condition (c) of Definition 4 demands a uniform bound on |Im(alpha_n)| for all zeros and condition (d) demands the sector condition |arg(alpha_n)| < pi/4; without the zero-free region, neither can be asserted for all zeros of zeta_M.","section":"Section 6"},{"comment":"The count 'In our case 31 zeros and a pole' and the subsequent computation of the zeros are described heuristically through an x-ray. This is not a certified argument-principle computation: there is no rigorous enclosure of the boundary integral, no error bound on the listed zeros, and no proof that the table contains every zero in the rectangle R=(-21,22)x(-10,80). A missed zero with small positive ordinate and large |Re(s)-1/2|, or a zero outside R with positive ordinate, would violate condition (a) or (d) of Definition 4. To support the claim that zeta_M gives a complex Riemann sequence, the authors must either provide a rigorous certificate of the zero count (for example interval arithmetic on the argument variation) or explicitly state that the example is only numerical evidence.","section":"Section 6, Figure 1 and zero table"},{"comment":"The paper's overarching claim that complex Riemann sequences exist is not established by Theorem 14 alone, because Theorem 14 constructs Riemann sequences without proving that their zeros are off the critical line. Remark 15 states 'This proves that there are complex Riemann sequences' after citing the Section 6 numerics. Since the Section 6 verification is presently non-rigorous, that sentence overreaches the proof. The manuscript should either upgrade the numerical verification to a rigorous one or weaken the claim to a conjecture or numerical demonstration, while Theorem 14 would then still establish the existence of Riemann sequences, but not their complexity.","section":"Remark 15 and Section 6"}],"minor_comments":[{"comment":"The abstract writes the displayed asymptotic relation for real x, while Theorem 3 states it for complex z in the sector |arg z| <= pi/2 - epsilon; please reconcile the notation so that the sector condition is visible from the outset.","section":"Abstract and Section 2"},{"comment":"In the displayed formula for zeta_M(s), the last term reads '(zeta(s,5/12)+zeta(7/12))'; the second Hurwitz zeta argument is missing an 's' and should be zeta(s,7/12).","section":"Section 6, definition of zeta_M"},{"comment":"The two-column presentation 'beta gamma beta gamma' is visually confusing; please use a proper table with column headers and clear row separators, and state explicitly the convention beta=Re(s), gamma=Im(s).","section":"Section 6, zero table"},{"comment":"The x-ray uses thick and thin lines to mark real and purely imaginary values; in grayscale or small print these may be hard to distinguish. Please add a legend and possibly labels for the axes.","section":"Section 6, Figure 1"},{"comment":"The statement says the function does not vanish 'nor on the square [-3/2,3/2] x [-3/2,3/2]'; this is a rectangle, not a square, and the wording could be clarified.","section":"Proposition 12"}],"recommendation":"major_revision","confidential_remarks":"The central structural result (Theorem 14) appears sound and is of genuine interest. The main obstacle to publication is that the existence of a complex Riemann sequence, a key advertised conclusion, currently rests on an unverified numerical computation. If the authors can supply rigorous error bounds for the zero-free region and the zero count, the paper would be suitable; otherwise the numerical example should be clearly labelled as evidence, and the claim of existence of complex Riemann sequences should be separated from the proved results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one for the idea, not for the Section 6 'proof'. The paper defines Riemann sequences — sequences satisfying the same asymptotic expansion as the τ_n from the zeta zeros — and proves there are infinitely many, all coming from crystalline measures via Meyer's earlier work. That is genuinely new. The derivation in Theorem 1 of the expanded product formula from the classical product is standard and looks correct; the coefficients are right. Theorem 14 is the real content: it shows that for odd N, T≥2, N^2>4NT+1, one can build zeta-like functions ζ_N whose zeros form a Riemann sequence. The argument uses the product representation (26), the zero-free region of Proposition 12, and the Hamburger-style functional equation. I checked the steps as far as I could; the logic is coherent. The paper is honest that the resulting sequences are probably not real, and the conjecture that the zeta zeros give the only real Riemann sequence is explicitly left open.\n\nThe soft spot is exactly where the stress-test note lands. Section 6 wants to exhibit a concrete complex Riemann sequence by writing down ζ_M and then asserting a zero-free half-plane for σ ≥ σ0 = 10.564... with no interval arithmetic or tail bound. The phrase 'where the dots represents all the other terms in ζ_M(s) taken with coefficients in absolute value' is not a bound; it is a promise that the tail is small. The x-ray in Figure 1 is good evidence but not a certified zero count. If a zero with Re α > 0 is missed, condition (a) or (d) of Definition 4 could fail. So the claim 'there exist non-real Riemann sequences' is not proven; it is supported by numerical evidence. That should be fixed or clearly downgraded in the paper.\n\nThe rest of the paper (Sections 7–8) derives properties of any Riemann sequence: an associated Z_α(s) with the expected poles and values. That part is mostly standard given the definition, and it is fine.\n\nWho should read it: people interested in the structure of zeta zeros and in Meyer's crystalline measure program. It deserves a serious referee — the main theorem is worth refereeing even if Section 6 needs revision. My recommendation: send it to review, ask for a rigorous certified computation of the example or an explicit caveat replacing 'we construct' with 'we exhibit numerical evidence for'.","headline":"The main existence theorem for Riemann sequences is analytically solid, but the paper's concrete complex example rests on a numerical x-ray, not a proof.","tokens_in":18659,"tokens_out":2859,"would_cite":true,"duration_ms":30078,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","52C23","30D99"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the zeta zeros' asymptotic expansion defines a whole family of 'Riemann sequences', including complex ones whose zeros lie off the critical line.","keywords":["Riemann zeta function","zeros of zeta","Riemann sequences","crystalline measures","functional equation","Hurwitz zeta function","asymptotic expansion","Riemann hypothesis"],"falsifier":"Run a rigorous contour count around the rectangle $R=(-21,22)\\times(-10,80)$ for $\\zeta_M$: a count other than 31 zeros plus one pole, or any zero with real part above $\\sigma_0=10.564029176912431172$, would disprove the concrete example; for the general theorem, a zero of some $\\zeta_N$ with $|\\operatorname{Im} t|\\ge 3/2$ for a small allowed $\\delta$ would break Proposition 12.","tokens_in":17662,"feed_emoji":"🔢","tokens_out":8639,"duration_ms":88031,"temperature":0.7,"pith_summary":"The paper tries to characterize the sequence of nontrivial zeros of the Riemann zeta function by an intrinsic property, without assuming the Riemann hypothesis. Its candidate is the 'Riemann sequence' property: an asymptotic expansion of $\\sum_k 2z/(z^2+\\alpha_k^2)$ with fixed coefficients $a_n$ built from Euler and Bernoulli numbers. The paper proves the zeta zeros form one such sequence, then constructs infinitely many others, and exhibits a concrete complex Riemann sequence whose computed zeros are not all on the critical line. If these constructions are right, the zeta zero sequence is not singled out by its asymptotic behaviour alone; the remaining conjecture is that it is the only real Riemann sequence. That would give a characterization of zeta's zeros that does not settle, and does not presuppose, the Riemann hypothesis.","feed_headline":"Zeta's zero pattern repeats in infinitely many new sequences","feed_subtitle":"The defining asymptotic law of zeta zeros holds for many other zero sets, including some with zeros off the critical line.","key_machinery":"The load-bearing object is the Riemann sequence (Definition 4): a sequence of complex numbers with positive real parts, bounded imaginary parts, and conjugate symmetry, satisfying the asymptotic expansion (16) with coefficients $a_{2n+1}=2^{-2n-2}(8-E_{2n})$ and $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n)$. The proof that the zeta zeros form such a sequence goes through the Weierstrass product for $\\Xi(t)$ and the Stirling expansion for $\\log\\Gamma$ (Theorems 1 and 3). For the new examples, the machinery is a construction of crystalline measures—tempered distributions whose support and Fourier support are both locally finite—via the finite Fourier transform on $\\mathbb{Z}/N^2\\mathbb{Z}$ (Propositions 8 and 9); applying the equivalence from [9] turns such a measure into a Dirichlet series $g_N(s)$ satisfying zeta's functional equation, and adding a small multiple of it to $\\zeta(s)$ yields $\\zeta_N(s)$ whose zeros are controlled by Proposition 12 and have the product representation (26). The explicit example $\\zeta_M(s)$ is a linear combination of Hurwitz zeta functions chosen from an explicit self-dual measure.","core_discovery":"On its own terms, the central discovery is that the zeta zeros satisfy the asymptotic law $$\\sum_{k\\in\\mathbb{N}}\\frac{2z}{$z^{2}$+\\$tau_k^{2}$} \\simeq \\frac12\\log\\frac{z}{2\\pi}+\\sum_{n=1}^\\infty \\frac{a_n}{z^n},$$ with $a_{2n+1}=2^{-2n-2}(8-E_{2n})$ and $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n)$, and that this law is not exclusive to them. Theorem 14 states that for every odd $N$ and integer $T\\ge 2$ with $N^2>4NT+1$, a small perturbation $\\zeta_N(s)=\\zeta(s)+\\delta g_N(s)$—where $g_N$ is an entire Dirichlet series satisfying the same functional equation as zeta—has a zero sequence that is again a Riemann sequence. Section 6 writes down an explicit $\\zeta_M(s)$ built from Hurwitz zeta functions whose computed zeros include points off the critical line, which the paper presents as a concrete complex Riemann sequence. The paper's conclusion, stated as a conjecture, is that the zeta zeros may be the only real Riemann sequence; if that uniqueness holds, it characterizes the zeta zeros intrinsically without settling the Riemann hypothesis.","pith_inferences":["Beyond the paper, the existence of complex Riemann sequences suggests that any characterization of zeta's zeros must add an arithmetic input—such as an Euler product—beyond analytic self-duality.","The construction makes a precise numerical prediction: a rigorous contour integral over the rectangle $R=(-21,22)\\times(-10,80)$, together with a verified zero-free bound $\\sigma\\ge\\sigma_0=10.564029176912431172$ for $\\zeta_M$, would turn the complex Riemann sequence example from numerical evidence into a theorem; the paper itself leaves that verification open.","Because the construction passes through crystalline measures, Riemann sequences can be viewed as the zeta-side of self-dual combs; a classification of self-dual measures with a gap at the origin would likely translate into a classification of all Riemann sequences, not just the real ones."],"forward_implications":["Every admissible pair $(N,T)$ with $N$ odd, $T\\ge 2$, and $N^2>4NT+1$ yields a meromorphic function $\\zeta_N$ sharing zeta's functional equation, with a unique simple pole at $s=1$, whose zeros with positive real part form a Riemann sequence; these functions are not expected to have Euler products.","The explicit $\\zeta_M$ provides a complex Riemann sequence, so the asymptotic property (16) alone does not force zeros onto the critical line; the Riemann hypothesis is not a consequence of this intrinsic zero law.","If the conjecture that only one real Riemann sequence exists is correct, then the zeta zero sequence is characterized among all Riemann sequences by being real—an intrinsic statement that bypasses the ordinary formulation of the Riemann hypothesis.","Every Riemann sequence gives a secondary zeta function $Z_\\alpha(s)$ with a double pole at $s=1$, simple poles at the odd negative integers, and prescribed values at the even negative integers (Theorems 26–27), so the family of sequences carries a whole class of zeta-like functions.","A real Riemann sequence different from $(\\tau_n)$ would be a concrete way to disprove the Riemann hypothesis without exhibiting a zero off the critical line; the paper explicitly points to this as a possible route."],"supporting_citations":[{"why":"supplies the crystalline-measure construction with a gap at the origin that Proposition 9 modifies.","marker":"[12]"},{"why":"provides the equivalence between self-dual crystalline measures and Dirichlet series satisfying zeta's functional equation used in Theorem 5.","marker":"[9]"},{"why":"gives the eigenvalue multiplicities of the finite Fourier transform on $\\mathbb{Z}/N^2\\mathbb{Z}$ used in Proposition 8.","marker":"[11]"},{"why":"supplies the asymptotic expansion of $\\log\\Gamma$ used in deriving the coefficient formula (5).","marker":"[10]"},{"why":"provides the Jensen-theorem bound on zero counting used to justify the product representation (26).","marker":"[3]"},{"why":"gives the Laplace-transform Tauberian theorem used to pass from (52) to the small-$x$ asymptotics (53).","marker":"[7]"}],"fun_headline_variants":["Zeta's zero law admits infinitely many other solutions","Riemann zero pattern not exclusive to zeta","Complex sequences mimic zeta zeros' asymptotic rule","Conjecture: only real Riemann sequences are zeta's","Zeta zeros' law extends to off-critical-line sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the infinite family, the proof depends on the product formula (26) holding exactly for the zeros of the constructed functions; for the concrete example, it depends on the numerical search having found every zero and on the claimed zero-free region being exact rather than approximate.","fun_headline_variants_meta":{"raw":{"variants":["Zeta's zero law admits infinitely many other solutions","Riemann zero pattern not exclusive to zeta","Complex sequences mimic zeta zeros' asymptotic rule","Conjecture: only real Riemann sequences are zeta's","Zeta zeros' law extends to off-critical-line sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1648,"prompt_tokens":1028,"completion_tokens":620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":543}},"tokens_in":644,"tokens_out":620,"duration_ms":6706,"temperature":1.0,"reasoning_tokens":543,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:47:06.294804+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a rigorous contour count around the rectangle $R=(-21,22)\\times(-10,80)$ for $\\zeta_M$: a count other than 31 zeros plus one pole, or any zero with real part above $\\sigma_0=10.564029176912431172$, would disprove the concrete example; for the general theorem, a zero of some $\\zeta_N$ with $|\\operatorname{Im} t|\\ge 3/2$ for a small allowed $\\delta$ would break Proposition 12.","supporting_citations":[{"cited_title":"Meyer , Measures with locally finite support and spectrum, Proc","cited_arxiv_id":null,"evidence_quote":"supplies the crystalline-measure construction with a gap at the origin that Proposition 9 modifies."},{"cited_title":"Hamburger , Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschenζ- Funktion äquivalent sind, Math","cited_arxiv_id":null,"evidence_quote":"provides the equivalence between self-dual crystalline measures and Dirichlet series satisfying zeta's functional equation used in Theorem 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the eigenvalue multiplicities of the finite Fourier transform on $\\mathbb{Z}/N^2\\mathbb{Z}$ used in Proposition 8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the asymptotic expansion of $\\log\\Gamma$ used in deriving the coefficient formula (5)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Jensen-theorem bound on zero counting used to justify the product representation (26)."},{"cited_title":"Doetsch, Handbuch der Laplace-Transformation, Band I, Theorie der Laplace-Transformation, Birkhäuser, Basel, 1950","cited_arxiv_id":null,"evidence_quote":"gives the Laplace-transform Tauberian theorem used to pass from (52) to the small-$x$ asymptotics (53)."}],"review_version":1}