{"id":"22a4b004-01b1-4f3b-ac4f-9b592df134e3","arxiv_id":"2607.20758","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The zero set of the Dirichlet series of even zeta values is fully classified: a zero-free right half-plane, one real zero, strip zeros obeying the Riemann–von Mangoldt law, and two conjugate left-plane strings.","lead":"William Banks gives a complete description of every zero of D(s)=Σζ(2n)n^{−s}, a Dirichlet series built from even zeta values that has no Euler product and no self-dual functional equation. The zeros split into four families, including two infinite strings in the left half-plane, and the count in the critical strip follows the same law as for ζ.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest_assumption identified Theorem 3.1 as the key premise, and I scrutinized exactly that and the dependent arguments. My conclusion is that the functional-equation premise is valid and the proof is internally consistent. The non-absolute convergence of the double series is handled appropriately by the k-grouping; the estimates in Lemma 5.4 are uniform and correct; the Rouché argument and the counting theorem are rigorous. Therefore no adjustment to the ACCEPT verdict is needed. The partial agreement reflects that the reader's chosen weakest point is indeed the foundational step, but I do not find it defective.","tokens_in":20816,"tokens_out":44604,"duration_ms":321860,"concrete_test":"Independently verify the functional equation (3.3) numerically: for s = -2-3i, -5+4i, and -10, compute the left side from D(s)=ζ(s)+E(s) with the series truncated at n=10^3, and compute the right side as Γ(1-s) times the k-grouped inner sums truncated at k=10^3, ℓ=10^3; require agreement to 10^{-10}. This directly tests the termwise-summation premise of Theorem 3.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The paper's central claim is a complete asymptotic classification of the zeros of D(s). The functional equation (Theorem 3.1) is the load-bearing premise; I checked the termwise summation over k and the subsequent analysis. The double series is not absolutely convergent, but Remark 3.2 correctly notes that convergence is via the k-grouped inner sums, each absolutely convergent for σ<0, and the values of the inner sums are absolutely summable in k since they are O(k^{-2}). Lemmas 5.2–5.4 are sound: the frequency comparison in Lemma 5.2 is exact, Lemma 5.3 follows directly from the functional equation, and Lemma 5.4's uniform bound (5.6) is proven with explicit constants and a positive η. The Rouché argument in Theorem 5.1 is legitimate: the disks are disjoint, the error bound (5.10) is uniform, and |h|/|g| decays like e^{-ηu_m/2}. The counting in Theorem 6.1 is standard and rigorous. No gap affecting the conclusions was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Dirichlet series D(s)=Σ_{n≥1} ζ(2n)n^{-s}, which continues meromorphically to C with a single simple pole at s=1. The central input is Theorem 3.1, an exact Hecke-type functional equation D(s)=Γ(1−s)Z(1−s) for σ<0, where Z(w) is a dual series over the complex logarithms of perfect squares, obtained from the Lipschitz summation formula with a mandatory grouping of terms (Remark 3.2). Using this functional equation, the author proves: zero-free half-plane σ≥σ0=1.500127440… (Prop. 4.1); a real zero ρ0=0.200411339… with D(σ)>0 for σ≤0 and σ>1, uniqueness conjectural (Prop. 4.2, Conj. 4.3); finiteness of zeros in every fixed left strip (Prop. 4.4); in the far left half-plane, all zeros are simple and form two conjugate strings satisfying ρ_m = 1 − iπ(2m+1)/L + O(e^{-κm}) with explicit L, Θ, κ (Theorem 5.1); and in the region −1/4≤σ≤σ0, zeros are a-points of ζ near a0=−(ζ(2)−1), with counting law N_D(T)=T/(2π)log(T/(2π))−T/(2π)+O(logT) (Theorem 6.1). No Riemann-type hypothesis is used.","tokens_in":21106,"tokens_out":22927,"duration_ms":164276,"significance":"If accepted, this is a rare complete asymptotic zero classification for a Dirichlet series that has neither an Euler product nor a self-dual functional equation. The functional equation is exact and elegant, and Riemann's functional equation appears as a single column of the dual series. The zero analysis is genuinely parameter-free: all constants σ0, L, Θ, η, κ are computed from definitions rather than fitted. The string theorem is quantitatively sharp, with geometrically decaying error, and the counting law is derived by a clean argument-principle argument. The numerical section is unusually thorough: argument-principle counts, Newton iteration, explicit working precision, and an independent mpmath recomputation. These strengths make the paper a valuable test case for the broader problem of zeros of non-Euler-product Dirichlet series.","major_comments":[],"minor_comments":[{"comment":"The proof uses the bound |ζ(−x)|≤1/2 on [0,1.3] without proof or citation. This is a real numerical input to the claim D(σ)>0 for σ≤0. Please supply a short justification (e.g., via the functional equation and monotonicity on that interval) or an explicit certified computation.","section":"§4, Proposition 4.2"},{"comment":"In the equality case, the sentence 'forces the phases n^{-it} to have the same value for all n≥2... occurs only if t=0' relies on the rational independence of log 2 and log(3/2). State or cite this elementary fact so the boundary case σ=σ0 is fully justified.","section":"§4, Proposition 4.1"},{"comment":"The phrase 'critical strip' is used for the region −1/4≤σ≤σ0, which extends beyond the usual critical strip 0<σ<1 and also below 0. The definition in Theorem 6.1 is explicit, but the terminology may mislead; consider calling it the 'central rectangle' or 'strip region'.","section":"§6"},{"comment":"The symbol L is used both for the complex constant L=log(π/log2)+iπ/2 and for its modulus |L|. This is notationally risky (e.g., in (5.1)–(5.2)); a separate symbol for the modulus would improve readability.","section":"§5, Theorem 5.1"},{"comment":"The claim of a 'complete unconditional description' is slightly stronger than what is proved, since uniqueness of the real zero is conjectural (Conjecture 4.3) and Levinson-type clustering is left open (Section 8). The abstract and Section 8 do qualify these points; a sentence in the introduction would further align the language with the proven results.","section":"§2 and §8"},{"comment":"The certified notebooks are 'available from the author on request'. For reproducibility, it would be preferable to make the code and printed output publicly available in a permanent repository.","section":"§7"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical content is sound and the numerical work is exemplary. The manuscript discloses the use of an AI assistant in the acknowledgments; this is transparent and does not affect my assessment. The only issues are small proof details and presentation points, all of which can be addressed locally."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a serious, carefully proved paper that delivers what the abstract promises — a complete asymptotic classification of the zeros of D(s)=Σ ζ(2n)n^{-s}, a Dirichlet series with neither an Euler product nor a self-dual functional equation. I agree with the reader's ACCEPT verdict; I found no load-bearing gap, and the stress-test note matches my own reading.\n\nWhat is actually new: the exact Hecke-type functional equation (Theorem 3.1), obtained by applying the Lipschitz summation formula to each polylog summand and summing over k, with the k=1 column reducing to Riemann's functional equation. The two-term reduction in Lemma 5.4 is the technical heart, and it is done honestly — the error is measured against the sum of the two leading terms, so either string frequency can dominate depending on the sector. The string theorem (Theorem 5.1) and the Riemann–von Mangoldt count (Theorem 6.1) follow from legitimate Rouché and argument-principle arguments. The numerics are certified by argument-principle counts with an independent mpmath check; the paper is clear about what is computed and what is proved.\n\nSoft spots, in proportion: they are minor. The bound |ζ(−x)|≤1/2 on [0,1.3] in Proposition 4.2 is asserted without proof; it is easy to verify, but it should be stated or referenced. Conjecture 4.3 (uniqueness of the real zero) is explicitly conjectural, and the paper treats it as such. The code is 'available on request' rather than deposited; that is a small but real inconvenience in 2026 for anyone who wants to reproduce the tables. The reliance on the Lipschitz summation formula for σ<0 in the stated form is standard; Remark 3.2 correctly explains why the double series must be grouped by k, and the grouping is justified by the absolute summability of the inner sums. I do not see a circular step anywhere: σ0, ρ0, L, Θ, η, κ are all computed from definitions, and the numerics check the asymptotics rather than set them.\n\nWho this is for: anyone working on zeros of Dirichlet series without Euler products, or on a-points of ζ; it is also a clean test case for the two-frequency interference heuristic. It deserves a serious referee — I would send it forward without hesitation. My only suggestions, besides fixing the small numerical inequality, are to deposit the notebooks and to add a sentence in Proposition 4.2 giving the source of the |ζ(−x)| bound.","headline":"A careful, genuinely new zero classification for a non-Euler-product Dirichlet series; the main theorems hold up, and only minor tidying is needed.","tokens_in":21491,"tokens_out":2895,"would_cite":true,"duration_ms":31567,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11M41"],"pacs":[],"model":"deepseek-v4-flash","headline":"The zero set of D(s)=Σ ζ(2n)n^{-s} is completely and unconditionally described: zero-free half-plane, a single real zero, Riemann-class strip zeros, and two simple left-half-plane strings.","keywords":["Dirichlet series","Riemann zeta function","functional equation","zeros of Dirichlet series","a-points","Lipschitz summation formula","polylogarithm","Hecke-type functional equation"],"falsifier":"Compute $D(s)$ to, say, 40 digits at a point with $\\sigma<0$ (e.g., $s=-5+10i$) using both the original Dirichlet series with the decomposition $D=\\zeta+E$ and the right-hand side $\\Gamma(1-s)Z(1-s)$ of the functional equation, truncating Z with a verified bound on the tail; agreement to the stated precision would support the identity, and any mismatch would refute the paper's central premise. Alternatively, search the left half-plane for a zero that violates the string asymptotic (5.2) or the spacing $2\\pi/|L|$.","tokens_in":20763,"feed_emoji":"🧮","tokens_out":5660,"duration_ms":40137,"temperature":0.7,"texified_at":"2026-08-05T21:36:47.895161+00:00","pith_summary":"This paper gives a complete unconditional description of the zero set of the Dirichlet series $D(s)=\\sum_{n\\ge1} \\zeta(2n) n^{-s}$, a series with positive real coefficients that has neither an Euler product nor a self-dual functional equation. Meromorphic continuation is immediate from $D=\\zeta+E$ with E entire, but locating the zeros is not; the paper proves an exact functional equation of Hecke type, $D(s)=\\Gamma(1-s)Z(1-s)$, whose dual series runs over complex logarithms of perfect squares, with Riemann's functional equation appearing as one column. That identity yields four families of zeros: a zero-free right half-plane, one real zero $\\rho_0=0.2004\\ldots$, strip zeros that are perturbed a-points of ζ and obey a Riemann–von Mangoldt law, and two complex-conjugate strings of simple left-plane zeros with explicit asymptotics. A sympathetic reader should care because complete zero-set classifications are extremely rare for Dirichlet series lacking Euler products and self-duality, and the proof is entirely unconditional.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8168,"prompt_tokens":1025,"completion_tokens":7143,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":1025,"completion_tokens_details":{"reasoning_tokens":6080}},"feed_headline":"Even-zeta Dirichlet series: every zero located, no RH assumed","feed_subtitle":"Zeros fall into four families: zero-free half-plane, one real zero, strip zeros, and two simple left-plane strings.","key_machinery":"The key object is the exact functional equation of Hecke type (Theorem 3.1), obtained by applying the Lipschitz summation formula to each summand in $D(s)=\\sum_k \\mathrm{Li}_s(k^{-2})$ and summing the identities over k. For $\\sigma<0$ it reads $D(s)=\\Gamma(1-s)Z(1-s)$, with $Z(w)=\\sum_{\\omega\\in\\Omega} \\omega^{-w}$ over the frequency set $\\Omega=\\{2\\log k + 2\\pi i \\ell : k\\in\\mathbb{N}, \\ell\\in\\mathbb{Z}\\} \\setminus \\{0\\}$, grouped by k because the double series is not absolutely convergent. In the left half-plane the proof reduces Z to its two leading frequencies — $\\omega_\\flat=2\\log 2$ contributed by the entire part $E(s)=\\sum(\\zeta(2n)-1)n^{-s}$ and $\\pm \\omega_\\sharp = \\pm 2\\pi i$ contributed by ζ itself — via a two-term approximation $Z(w)=\\omega_\\flat^{-w}+\\omega_\\sharp^{-w}+O(e^{-\\eta u}(|\\omega_\\flat^{-w}|+|\\omega_\\sharp^{-w}|))$. The constant $L=\\log(\\omega_\\sharp/\\omega_\\flat)=\\log(\\pi/\\log 2)+i\\pi/$","core_discovery":"The central claim is that the zero set of $D(s)=\\sum \\zeta(2n) n^{-s}$ is completely and unconditionally described. Using the Lipschitz summation formula termwise in the polylogarithm decomposition, the paper establishes an exact functional equation $D(s)=\\Gamma(1-s)Z(1-s)$ for $\\sigma<0$, where the dual series $Z(w)=\\sum_{\\omega\\in\\Omega} \\omega^{-w}$ runs over the set $\\Omega=\\{\\log k^2 + 2\\pi i \\ell : k\\in\\mathbb{N}, \\ell\\in\\mathbb{Z}\\} \\setminus \\{0\\}$ with a mandatory grouping of terms; the k=1 column is precisely Riemann's functional equation. Interference between the two smallest frequencies, $\\omega_\\flat=2\\log 2$ from the entire part E and $\\pm \\omega_\\sharp = \\pm 2\\pi i$ from ζ, controls every zero of large modulus in the left half-plane. The paper proves that D is zero-free for $\\sigma \\ge \\sigma_0 = 1.500127440\\ldots$, that D has a r","pith_inferences":["The two-frequency interference picture should extend to the family D_α(s)=Σ ζ(αn)n^{-s} for real α>1: for α<2π/log 2 the same strings are predicted (the paper sketches this in Section 8); if correct, the entire one-parameter family would have a uniform zero-set description.","The author leaves open whether almost all strip zeros cluster near σ=1/2 (a Levinson-type statement); the bounded, nearly constant perturbation −E(s) makes a proof plausible by adapting mean-value estimates for ζ.","If the real zero ρ0 is indeed unique, a rigorous monotonicity proof of D on (0,1) would follow from a moderate amount of numerical verification of E′(σ) bounds; this is a concrete testable extension.","The method locates the left-plane zeros by Rouché's theorem around the two-term approximation; one can test the sharp constant in the error term by computing the distance from actual zeros to the predicted points for m up to a few hundred."],"forward_implications":["For the series D, the full zero set is known without any unproved hypothesis: zero-free right half-plane, one real zero, a Riemann–von Mangoldt strip count, and two simple left-plane strings.","The strip zeros of D mirror the a-points of ζ for a near −(ζ(2)−1), so the value-distribution theory of ζ transfers to D at this quantitative level.","Riemann's functional equation appearing as a single column of the dual series suggests a new structural family of Hecke-type equations generated by special values of ζ.","The counting law N_D(T) agrees to leading order with that of ζ itself, despite D having neither Euler product nor self-duality, so zero counting alone does not detect those properties.","The exponentially decaying error in the string asymptotics means the left-plane zeros are eventually governed by a finite two-term exponential-sum model, making them computable to high precision."],"fun_headline_variants":["Complete zero map for Dirichlet series of even zeta values","Four families of zeros for even-zeta Dirichlet series","Zero set fully characterized for even-zeta Dirichlet series","Hecke-type equation yields complete zero description","No RH needed: all zeros of even-zeta D(s) found"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire zero analysis rests on the classical Lipschitz summation formula being applied termwise to the polylogarithm sum and the resulting identities being summed over k; if that functional equation $D(s)=\\Gamma(1-s)Z(1-s)$ were invalid for $\\sigma<0$, every subsequent zero-location argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Complete zero map for Dirichlet series of even zeta values","Four families of zeros for even-zeta Dirichlet series","Zero set fully characterized for even-zeta Dirichlet series","Hecke-type equation yields complete zero description","No RH needed: all zeros of even-zeta D(s) found"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000936,"raw_usage":{"total_tokens":3930,"prompt_tokens":920,"completion_tokens":3010,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":2928}},"tokens_in":664,"tokens_out":3010,"duration_ms":18607,"temperature":1.0,"reasoning_tokens":2928,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:29:50.666417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $D(s)$ to, say, 40 digits at a point with $\\sigma<0$ (e.g., $s=-5+10i$) using both the original Dirichlet series with the decomposition $D=\\zeta+E$ and the right-hand side $\\Gamma(1-s)Z(1-s)$ of the functional equation, truncating Z with a verified bound on the tail; agreement to the stated precision would support the identity, and any mismatch would refute the paper's central premise. Alternatively, search the left half-plane for a zero that violates the string asymptotic (5.2) or the spacing $2\\pi/|L|$.","supporting_citations":[],"review_version":1}