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REVIEW 5 major objections 4 minor 15 references

The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A sincere but currently broken adaptation of Pintz's method; two load-bearing gaps break the converse. read the letter →

arxiv 2505.23795 v1 pith:SYWDYWAX submitted 2025-05-26 math.NT

Songlin Han This is my paper · ORCID
classification math.NT MSC 11M2611N0511P32
keywords Riemannzetafunctionzero-freeregionsmoothweightedprimenumbertheoremerrortermGoldbachrepresentationsvonMangoldtpower-sumlower-boundmethod
topics Goldbach's ConjectureThe Riemann HypothesisThe Twin Prime Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

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.

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 (5)
  1. [Section 2, Eq. (16)] 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.
  2. [Section 2, after Eq. (14)] 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.
  3. [Section 4.1, near Eq. (22)] 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.
  4. [Section 4.2, Eq. (23)] 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.
  5. [Section 3.2, Lemma 2.2] 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.
minor comments (4)
  1. [Section 2, Theorem 1.2 proof] 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.
  2. [Section 2, Theorem 1.3] 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.
  3. [Throughout] 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.
  4. [Section 4.1, paragraph after equation (22)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified; the main flagged issues are correctness gaps, not definitional reductions.

full rationale

The derivation is not circular. Theorem 1.1 is proved from the Hardy–Littlewood explicit formula (3), an upper-bound lemma on W(x) (Lemma 2.2), and a lower-bound lemma on D(x) (Lemma 2.3) adapted from Pintz’s method; these are external results, not assumptions of the conclusion. The zero-free-region hypothesis enters the first part only through a comparison of the quantities ω_η(x) and ω(x), and the converse uses the assumed Δ-bound at x0 to contradict an unconditional lower bound obtained from Lemma 2.3; neither direction defines its target quantity into existence. The Goldbach theorems are applications of Theorem 1.1 rather than renamings of the input. The paper contains no data fitting, no fitted parameter renamed as a prediction, and no load-bearing citation by the present author; references [1], [3], [14], and [15] are prior work by other researchers. The proof gaps highlighted by a correctness review, especially the use of Lemma 2.3’s average lower bound as a pointwise lower bound on |Δ(x0)| at equation (16), would make the converse proof incomplete if confirmed, but an invalid inferential step is not circular: it does not make the theorem’s conclusion equivalent to its premise by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or invented entities. Its central claims rest on standard explicit formulas, on unproved borrowings from Pintz's work (Lemmas 2.2 and the Tura'n bound), and on an unjustified conversion from an average to a pointwise bound in the converse proof. These are the load-bearing dependencies.

assumptions (5)
  • standard math Explicit formula (3): Psi(x) = x - sum_rho Gamma(rho) x^rho - log 2pi + O(1/x)
    Attributed to Hardy and Littlewood, used as the starting point for the error analysis.
  • ad hoc to paper Pintz's estimates (4.9)-(4.12) in [15] apply verbatim to the Gamma(rho)-weighted sum W(x)
    Lemma 2.2 is stated without proof, citing Pintz's equations directly.
  • ad hoc to paper The Tura'n power sum lower bound E(mu) >= exp(-epsilon1 omega) carries over from [14] to this setting
    Section 4.2 asserts the bound without verifying hypotheses.
  • ad hoc to paper The average lower bound D(x) can be converted to a pointwise lower bound at x0
    Used in the converse proof of Theorem 1.1, equation (16); not justified.
  • domain assumption Classical zero-free region for zeta(s) (no zeros on sigma >= 1)
    Used in Section 4.1 to lower-bound omega(x).

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Cite this review

Pith. "Pith review of The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function." pith.science (2026). https://pith.science/paper/SYWDYWAX

@misc{pith2026250523795,
  author       = {Pith},
  title        = {Pith review of: The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYWDYWAX}},
  note         = {Machine review of arXiv:2505.23795}
}
read the original abstract

We study the error bound for a smooth weighted prime number theorem, and its implication to the zero-free region for the Riemann zeta function using the method of Pintz. We also give an application to the average number of smooth weighted Goldbach representations and generalize the result to the case of smooth weighted average k-Goldbach representations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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