{"id":"b58eca53-b63e-45f7-8d9b-6a41f4af977f","arxiv_id":"2505.23795","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper attempts to prove an equivalence between the error in a smooth weighted prime number theorem and zero-free regions for the Riemann zeta function, but critical proof errors undermine the result.","lead":"This paper claims that the error in a smoothly weighted prime counting formula is equivalent to where the Riemann zeta function has no zeros, following a method of Pintz, with applications to Goldbach-type averages. The proof contains a false gamma-function estimate and a gap between average and pointwise error bounds, so the claims are not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse of Theorem 1.1 is unsupported: (16) converts Lemma 2.3's average lower bound into a pointwise lower bound on |Δ(x0)| without justification; the claimed equivalence does not follow as written.","rationale":"I read the paper as attempting to prove a smooth-weighted analogue of the Ingham–Pintz equivalence, with Theorem 1.1 as the central claim. The upper-bound direction is standard modulo notation. The converse is the critical direction: it must rule out a zero ρ0 from a pointwise upper bound on Δ. Lemma 2.3 is the only lower-bound tool, and it produces a lower bound on the average D(x). The manuscript's line (16) uses Lemma 2.3 as if it gave a pointwise lower bound on |Δ(x0)|. This is not a matter of constants; it is a category error. The reader's weakest_assumption identifies exactly this gap, so I agree with the reader's assessment. I also verified the Section 4.1 issue: the assertion |1+β0−ε1/2| ≍ ε1^{-1} is false because β0 is near 1, making the gamma argument near 2; the cited reflection formula is also not the standard one. That is a genuine error in the proof of Lemma 2.3, but I do not treat it as the single most load-bearing concern because the lemma's statement may still be salvageable through Pintz's method. The average-to-pointwise gap sits directly inside the proof of Theorem 1.1's converse and cannot be patched by the text as written. Therefore the central claim is not established, and the reader's REJECT verdict stands with no adjustment.","tokens_in":12225,"tokens_out":15382,"duration_ms":156958,"concrete_test":"Re-derive the contradiction in Section 2 in terms of D(x0): use (12) to bound D(x0) above by (1/x0)∫_0^{x0} Cε u exp(-(1-ε)ϖ(u)) du plus a negligible initial segment, and use Lemma 2.3 to bound D(x0) below by (x0^{δ0}γ0)^{-ε} x0^{β0}/γ0 at x0 = exp(logγ0/η(logγ0)). Compare the logarithmic exponents over all admissible η with η'→0 and δ0<η(logγ0). If some admissible η makes the averaged lower bound weaker than the averaged upper bound, the converse is genuinely false; if the exponents always conflict, the argument is repairable, but (16) still needs a separate pointwise derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2's proof of the converse of Theorem 1.1 breaks at (16). Lemma 2.3 provides a lower bound only for the average D(x0) = (1/x0)∫_0^{x0} |Δ(u)| du, namely D(x0) ≥ (x0^{δ0} γ0)^{-ε} x0^{β0}/γ0. The proof then asserts, 'By Lemma 2.3 ... we obtain |Δ(x0)| ≥ ...' and uses this pointwise lower bound in the contradiction with the pointwise upper bound (12). No step converts an average lower bound into a pointwise lower bound at the chosen x0; a function can have a large average yet be arbitrarily small at an individual point. Thus the contradiction does not follow as written. A repair is not automatic: replacing (16) by a bound on D(x0) would require integrating (12), and the manuscript supplies no such argument. Moreover, the choice x0 = exp(log γ0/η(log γ0)) is not checked against Lemma 2.3's range hypotheses x0 > γ0^{1/ε^{10}} and (log x0)^{1/2} > ε^{-10}. Since this step is exactly what establishes the converse direction, the central equivalence is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the error term Δ(x) of the smooth weighted prime number formula Δ(x) = Σ_n (Λ(n)-1)e^{-n/x}. It claims an equivalence between upper bounds for |Δ(x)| and zero-free regions for ζ(s) (Theorem 1.1), following the method of Pintz. It then derives analogous statements for smooth weighted Goldbach representations (Theorems 1.2 and 1.3). The proof relies on an explicit formula for Δ(x) as a sum over zeta zeros, an upper bound for the weighted sum W(x), and a lower bound for the average D(x) obtained via Turán's power sum method. The central claim is the converse direction of Theorem 1.1, which would recover a zero-free region from the assumed error bound.","tokens_in":12570,"tokens_out":4254,"duration_ms":48712,"significance":"If established, the equivalence in Theorem 1.1 would be a natural smooth-weight analogue of the classical Ingham–Turán–Pintz theorem and would also yield the advertised applications to average Goldbach representations and k-Goldbach representations. The paper correctly identifies the explicit formula and the potential utility of Pintz's method as the right tools. However, the proof as written has load-bearing gaps: one inequality at equation (16) converts an average lower bound into a pointwise lower bound without justification, and a key gamma-function estimate in Section 4.1 is numerically false. The cited external estimates from Pintz are also transferred to a Gamma-weighted sum without a proof that the transfer is valid. These are not cosmetic issues; they affect the main equivalence, so the current manuscript cannot be accepted.","major_comments":[{"comment":"The converse direction of Theorem 1.1 breaks at equation (16). Lemma 2.3 supplies a lower bound for the average D(x0) = (1/x0)∫_0^{x0} |Δ(u)|du, not for the pointwise value |Δ(x0)|. The text asserts 'By Lemma 2.3 ... we obtain |Δ(x0)| ≥ ...' and then uses this pointwise lower bound in the contradiction with the pointwise upper bound (12). A positive average does not imply a pointwise lower bound at the chosen point x0. The manuscript provides no argument (e.g., an integrated upper bound over a short interval) that would justify such a conversion. This gap is load-bearing because it is exactly the step that establishes the converse direction of Theorem 1.1.","section":"Section 2, Eq. (16)"},{"comment":"The choice x0 = exp(log γ0 / η(log γ0)) is not checked against the hypotheses of Lemma 2.3, which require x0 > γ0^{1/ε^{10}} and (log x0)^{1/2} > ε^{-10}. Since Lemma 2.3 is the only source of the lower bound used in (16), omitting this verification leaves the contradiction argument incomplete even apart from the pointwise-versus-average issue.","section":"Section 2, after Eq. (14)"},{"comment":"The bound for U(0,1] relies on the assertion that |1+β0 - ε1/2| ≍ ε1^{-1}. This is false: since β0 = 1 - δ0 with δ0 < ε^{10}, the argument in question is 2 - δ0 - ε1/2, which is near 2 and bounded away from 0. Consequently the claimed decay Γ(1+β0 - ε1/2) ≪ exp(-c ε1^{-1} log(1/ε1)) does not follow from Stirling's formula and the reflection formula. The conclusion U(0,1] ≪ exp(-ω) is therefore unsupported, and this region is an essential part of the lower-bound proof for Lemma 2.4.","section":"Section 4.1, near Eq. (22)"},{"comment":"The Turán power sum lower bound E(μ) ≥ exp(-ε1 ω) is transferred from Pintz [14] with the comment that the factor ρΓ(ρ) 'essentially changes nothing' in the narrow rectangular region. This is not demonstrated. In Pintz's setting the relevant sum has coefficients of comparable size and a known structure; here the coefficients ρΓ(ρ) vary with both real and imaginary parts, and the number of zeros in the rectangle is not controlled in a way that makes the transfer immediate. No proof of (23) in the Gamma-weighted case is supplied. Since (23) is the decisive lower bound behind Lemma 2.4 and hence behind the converse direction of Theorem 1.1, this is a second load-bearing gap.","section":"Section 4.2, Eq. (23)"},{"comment":"The upper bound for W(x) is obtained by saying that Pintz's estimates (4.9)-(4.12) of [15] can be used 'directly' because the gamma function is bounded by a constant less than 1. This is not a direct consequence: the classical estimates in [15] are for sums of x^ρ/ρ without a Γ(ρ+1) factor. A global bound of the gamma factor by a constant less than 1 is also not proved for all nontrivial zeros, and even if such a bound held, the more delicate exponential-sum structure of the estimates would need to be re-examined. As written, Lemma 2.2, which drives the first direction of Theorem 1.1, is not fully established.","section":"Section 3.2, Lemma 2.2"}],"minor_comments":[{"comment":"The passage from (18) to (19) uses a squaring of the explicit formula and then a square-root of the error term; this is plausible but should be written with care about the implied constants and the fact that the sum over ρ is conditionally convergent. A short derivation would improve readability.","section":"Section 2, Theorem 1.2 proof"},{"comment":"The proof of Theorem 1.3 is very terse: the 'binomial theorem' step and the passage from F_k(x) to (Ψ(x)-x)^k are not fully explained, and the displayed error term x^{k-1} appears without derivation. This application is not the main focus, but it should be made precise if kept.","section":"Section 2, Theorem 1.3"},{"comment":"There are several minor typographical and reference inconsistencies, such as referencing 'Theorem 2' when Theorem C is meant, and the spelling 'Tura'n' instead of 'Turán'. These do not affect the mathematics but should be corrected.","section":"Throughout"},{"comment":"The sentence 'The first factor is decreasing rapidly (at least exp(-2ω)) but the second term only contributes at most exp(ω)' is hard to parse; the role of the two factors and the reason the product is exp(-ω) should be stated explicitly.","section":"Section 4.1, paragraph after equation (22)"}],"recommendation":"reject","confidential_remarks":"The manuscript promises a complete proof of a smooth-weight version of the Pintz equivalence, but the arguments at equations (16), (22), and (23) are exactly the points where the Gamma weight changes the classical calculation, and in all three places the paper substitutes assertion for proof. The average-to-pointwise issue alone is fatal for the converse direction, and the false gamma-reflection estimate in §4.1 is a concrete error, not a mere gap. I do not see a local repair that would fit within the current manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know two things about this paper before deciding whether to spend time on it. The intended result is exactly the right kind of thing to want: a smooth-weight analogue of Pintz's equivalence between the PNT error term and zero-free regions, plus a proof of the converse to the recent Billington et al. Goldbach theorem. But as written, the proof of the converse direction does not hold. There are two load-bearing errors.\n\nThe first is in Section 2, equation (16). Lemma 2.3 gives a lower bound on the average D(x) = (1/x)∫_0^x |Δ(u)|du, not on the pointwise value |Δ(x0)|. The proof then asserts a pointwise lower bound at the chosen x0. Nothing in the paper converts an average bound into a pointwise one, and this step is exactly what produces the contradiction with the pointwise upper bound (12). The converse of Theorem 1.1 is therefore unsupported.\n\nThe second is in Section 4.1. The claim that |1+β0−ε1/2| ≍ ε1^{-1} is false: β0 = 1−δ0 with δ0 small, so the gamma argument is near 2, not near ε1^{-1}. The exponential decay in the bound for U(0,1] depends on this, so the proof of Lemma 2.4 collapses at that point.\n\nThere are also smaller issues: the key lower bound for E(μ) is quoted from Pintz [14] without showing the adaptation works, and Theorem 1.3's proof is a sketch.\n\nWhat the paper does well: the explicit formula manipulation is standard and the overall structure is a faithful, clearly presented imitation of Pintz's approach. The intended results are natural, and the paper is honest about following Pintz. If the gaps were repaired, the smooth-weight equivalence would be a useful addition to the literature.\n\nWho is this for? Analytic number theorists who work on Pintz's method, the PNT error term, and zero-free regions. It deserves a serious referee because the questions are legitimate and the flaws are specific and repairable in principle. But the current manuscript does not establish its theorems. I would send it to review, with the expectation of major revision. In its present form, I would not accept it.\n\nBest,\n[Your name]","headline":"A sincere but currently broken adaptation of Pintz's method; two load-bearing gaps break the converse.","tokens_in":13019,"tokens_out":4886,"would_cite":false,"duration_ms":44818,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N05","11P32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The central claim is that a smooth weighted prime error term and the zero-free region of the Riemann zeta function are equivalent, with analogues for Goldbach and k-Goldbach averages.","keywords":["Riemann zeta function","zero-free region","smooth weighted prime number theorem","error term","Goldbach representations","von Mangoldt function","power-sum lower-bound method"],"falsifier":"Compare the average lower bound $D(x_0)=x_0^{-1}\\int_0^{x_0}|\\Delta(u)|\\,du$ from Lemma 2.3 with the pointwise value $|\\Delta(x_0)|$ at $x_0=\\exp(\\log\\gamma_0/\\eta(\\log\\gamma_0))$ for an explicit zero $\\rho_0=1-\\delta_0+i\\gamma_0$ and a slowly varying $\\eta$. The converse proof requires the pointwise value to inherit the average lower bound, and this is directly checkable from the explicit formula summed over zeros up to a finite height.","tokens_in":1960,"feed_emoji":"🔢","tokens_out":5947,"duration_ms":138350,"temperature":0.7,"pith_summary":"This paper studies the error term in a smoothly weighted prime number theorem: the difference between the exponentially weighted sum of the von Mangoldt function and the smooth baseline. Its central claim is that an assumed zero-free region for the Riemann zeta function forces this weighted error to decay at a specific exponential rate, and, conversely, that a sufficiently strong decay of this weighted error forces the same zero-free region. If the claim is correct, the smooth weighted prime formula and the location of zeta zeros become interchangeable, just as in the classical prime number theorem. The paper also derives matching equivalences for smoothed average Goldbach representations and k-term Goldbach representations.","feed_headline":"Weighted prime error term is equivalent to zeta zero-free region","feed_subtitle":"If the claim holds, a sharp bound on this smooth error forces a zero-free region, and conversely.","key_machinery":"The object carrying the argument is the smooth weighted error $\\Delta(x)=\\sum_n(\\Lambda(n)-1)e^{-n/x}$, together with its average $D(x)=x^{-1}\\int_0^x|\\Delta(u)|\\,du$, its pointwise maximum $S(x)=\\max_{u\\leq x}|\\Delta(u)|$, and the zero sum $W(x)=\\sum_{|\\gamma|\\leq x}|\\Gamma(\\rho+1)|x^\\beta/|\\gamma|$. The paper establishes that their logarithmic growth rates are all equivalent, $\\omega(x)\\sim\\omega_D(x)\\sim\\omega_S(x)\\sim\\omega_W(x)$, where $\\omega(x)$ is the infimum over zeros of $\\delta\\log x+\\log\\gamma$. This equivalence is proved by upper-bounding $W(x)$ through the exponential decay of the gamma function and lower-bounding $D(x)$ through a power-sum argument over a narrow rectangle of zeros, followed by a residue computation of a Gaussian-smoothed Mellin transform. Once the growth rates are matched, the machinery interpolates between arbitrary zero-free regions and arbitrary error bounds.","core_discovery":"Theorem 1.1 states the central equivalence. If $\\zeta(\\sigma+it)\\neq 0$ for $\\sigma>1-\\eta(\\log|t|)$, then the smooth weighted error $\\Delta(x)=\\sum_n(\\Lambda(n)-1)e^{-n/x}$ satisfies $\\Delta(x)\\ll x\\exp(-(1-\\epsilon)\\omega_\\eta(x))$, where $\\omega_\\eta(x)=\\inf_{t\\geq 1}(\\eta(t)\\log x+\\log t)$. Conversely, if $\\Delta(x)\\ll x\\exp(-(1-\\epsilon)\\varpi(x))$ with $\\varpi(x)=\\min_{u\\geq 0}(\\eta(u)\\log x+u)$ and $\\eta'(u)\\to 0$, then $\\zeta(\\sigma+it)\\neq 0$ for $\\sigma>1-\\eta(\\log|t|)$ and all sufficiently large $|t|$. The proof mirrors the classical equivalence for the partial-sum prime error, replacing the term $x^\\rho/\\rho$ with $\\Gamma(\\rho)x^\\rho$ and using a power-sum lower-bound argument to show that the average of the weighted error is dominated by the contribution of the zeros closest to the line $\\sigma=1$. The same mechanism yields the Goldbach and k-Goldbach analogues in Theorems 1.2 and 1.3.","pith_inferences":["If the equivalence is made fully effective, a finite computation using the first many zeta zeros and an explicit $\\eta$ could turn a numerical zero-free region into a numerical bound on the smooth weighted error and back.","The same smoothing construction should transfer to other L-functions and automorphic forms, wherever an explicit formula with a gamma factor and a zero-counting estimate is available.","A sharper conversion from the average lower bound $D(x)$ to a pointwise lower bound would let average Goldbach statistics alone control zero-free regions, without requiring a separate pointwise lower bound for $\\Delta$.","For k-Goldbach sums, the equivalence suggests that higher moments of the smoothing could carry independent constraints on zeta zeros, not just the first moment used here."],"forward_implications":["A zero-free region of width $\\eta(\\log|t|)$ forces $\\Delta(x)\\ll x\\exp(-(1-\\epsilon)\\inf_{t\\geq 1}(\\eta(t)\\log x+\\log t))$.","Conversely, if $\\Delta(x)\\ll x\\exp(-(1-\\epsilon)\\min_{u\\geq 0}(\\eta(u)\\log x+u))$ with $\\eta'(u)\\to 0$, then $\\zeta$ has no zeros in $\\sigma>1-\\eta(\\log|t|)$ for large $|t|$.","If the smooth Goldbach average satisfies $\\sum_n\\psi_2(n)e^{-n/x}=x^2+O(x^{2-2\\eta(\\log x)})$, then the zero-free region $\\sigma>1-\\eta(\\log|t|)$ follows.","For each integer $k\\geq 1$, the k-fold smoothed Goldbach sum $F_k(x)$ obeys the analogous equivalence: a zero-free region implies $F_k(x)=x^k+O(x^{k-\\eta(\\log x)})$, and the error bound $x^k+O(x^{k-k\\eta(\\log x)})$ implies the zero-free region."],"supporting_citations":[{"why":"Supplies the framework this paper follows: comparing error-term oscillations with zero-free regions through the sizes of zero sums.","marker":"[15]"},{"why":"Provides the power-sum lower-bound method used to show the average error is at least the dominant zero contribution.","marker":"[14]"},{"why":"Gives the explicit formula for the smooth weighted prime sum and the conditional bound $\\Delta(x)=O(\\sqrt{x})$ under the Riemann hypothesis.","marker":"[5]"},{"why":"States the smooth Goldbach average estimate and the unproved converse that Theorem 1.2 supplies.","marker":"[1]"},{"why":"Supplies the equivalence between the classical prime number theorem and its smooth weighted form used as Proposition 2.1.","marker":"[4]"},{"why":"Provides the contour integral identity linking Goldbach averages to the smooth weighted error.","marker":"[3]"},{"why":"Is the classical reference for the prime number theorem error term and zero-free-region equivalence being adapted.","marker":"[7]"},{"why":"Contains the classical lower-bound method for the partial-sum error that the weighted version extends.","marker":"[12]"}],"fun_headline_variants":["Smooth prime error term equals zeta zero-free region","Weighted prime error implies zero-free region and vice versa","Pintz method: smooth prime error links zeta zeros","Error in smooth weighted prime formula mirrors zeta gaps","Prime error bound and zeta zero-free region: equivalence"],"cache_read_input_tokens":15104,"weakest_assumption_plain":"The converse proof assumes that a lower bound on the average size of the weighted error on the interval $[0,x_0]$ implies the same lower bound at the single point $x_0$ where the contradiction is evaluated.","fun_headline_variants_meta":{"raw":{"variants":["Smooth prime error term equals zeta zero-free region","Weighted prime error implies zero-free region and vice versa","Pintz method: smooth prime error links zeta zeros","Error in smooth weighted prime formula mirrors zeta gaps","Prime error bound and zeta zero-free region: equivalence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1237,"prompt_tokens":864,"completion_tokens":373,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":292}},"tokens_in":480,"tokens_out":373,"duration_ms":3779,"temperature":1.0,"reasoning_tokens":292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:05:05.972916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the average lower bound $D(x_0)=x_0^{-1}\\int_0^{x_0}|\\Delta(u)|\\,du$ from Lemma 2.3 with the pointwise value $|\\Delta(x_0)|$ at $x_0=\\exp(\\log\\gamma_0/\\eta(\\log\\gamma_0))$ for an explicit zero $\\rho_0=1-\\delta_0+i\\gamma_0$ and a slowly varying $\\eta$. The converse proof requires the pointwise value to inherit the average lower bound, and this is directly checkable from the explicit formula summed over zeros up to a finite height.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the framework this paper follows: comparing error-term oscillations with zero-free regions through the sizes of zero sums."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the power-sum lower-bound method used to show the average error is at least the dominant zero contribution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the explicit formula for the smooth weighted prime sum and the conditional bound $\\Delta(x)=O(\\sqrt{x})$ under the Riemann hypothesis."},{"cited_title":"The av- erage number of Goldbach representations and zero-free region s of the Riemann zeta function","cited_arxiv_id":null,"evidence_quote":"States the smooth Goldbach average estimate and the unproved converse that Theorem 1.2 supplies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between the classical prime number theorem and its smooth weighted form used as Proposition 2.1."},{"cited_title":"Goldston and Ade Irma Suriajaya","cited_arxiv_id":null,"evidence_quote":"Provides the contour integral identity linking Goldbach averages to the smooth weighted error."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the classical reference for the prime number theorem error term and zero-free-region equivalence being adapted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the classical lower-bound method for the partial-sum error that the weighted version extends."}],"review_version":1}