Explicit bounds for Buchstab's function

Andreas Weingartner

REVIEW 4 major objections 5 minor 10 references

Explicit bounds for Buchstab's function

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes rigorous closed-form bounds for Buchstab's function ω(u): for every u≥3, the difference W(u)=ω(u)−e^{−γ} is captured by an explicit cosine formula whose error is bounded by simple powers of u, with no numerical soluti

desk verdict Explicit, easy-to-evaluate bounds for Buchstab's function with a genuine new error term — but the full-range theorems rest on numerical checks that are described, not supplied. read the letter →

arxiv 2607.21883 v2 pith:AMECFK2B submitted 2026-07-24 math.NT

Andreas Weingartner This is my paper · ORCID

classification math.NT MSC 11N2511N0511N3541A60

keywords Buchstab'sfunctiondelaydifferentialequationsaddlepointmethodroughnumbersincompletegammaLambertWexpliciterrorboundsprimesinshortintervals

topics The 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

The paper sets out to make the classic asymptotic description of Buchstab's function ω(u)—the function that counts how common integers are that have no small prime factors—fully explicit. It proves several versions of the identity W(u)=ω(u)−e^{−γ}=2|Φ(u)|cos(arg Φ(u))+error, where Φ(u) is built from the incomplete gamma function and a saddle point ζ(u) that solves e^ζ=−uζ. Each version comes with a rigorous, easily computable error bound, the sharpest being |error|<420 u^{−6} on 6≤u≤19 and <0.005 u^{−2} on u≥16. Because ω(u) oscillates about the limiting value e^{−γ}, these bounds directly quantify the fluctuations that drive the known results on primes in short intervals. A sympathetic reader would care because this replaces the need to solve a delay differential equation numerically with a few evaluations of standard special functions.

What carries the argument

The central object is the saddle point ζ(u), the unique solution in the strip μ≥1, 0<η<3π/2 of e^ζ=−uζ, equivalently ζ=−W_1(1/u). Near this point the Laplace integrand e^{J(s)+us} has a vanishing derivative, so the main term Φ(u)=exp{−uζ+J(−ζ)}/√(2πu(1−1/ζ)) captures the oscillatory structure of W(u). The proof couples this with the identity J(s)=Γ(0,s), the derivative formula J^{(k)}(s)=(−1)^k e^{−s}/s times an explicit polynomial, Taylor polynomials of orders three and seven, and tail bounds showing the integrand decays like e^{−c u τ^2}, e^{−c u}, or e^{−8u/log^2 u} away from the saddle point. The explicit correction terms α(u) and β(u) are rational functions of ζ that give the next terms

What would settle it

Take u=1000, compute the interval from Theorem 2 (the main term plus the claimed |θ_2|<0.005·10^{−6}=5×10^{−6}) and independently solve the delay equation with a high-precision solver at u=1000; if the value falls outside the interval, the theorem's error bound is false. Equally, one of the finite-range inequalities—such as the claimed maximum of Re(H(u,τ))(log^2 u)/u over 50≤u≤10^4, 4≤τ≤370 being below −8—can be checked by an independent numerical integration; a single violation refutes the corresponding bound.

Watch

Extended reading notes

Core claim

The paper proves that W(u) equals 2|Φ_2(u)|cos(arg Φ_2(u)) plus an explicit error θ_2(u), where Φ_2(u)=Φ(u)(1+α(u)), Φ(u)=exp{−uζ+J(−ζ)} divided by the square root of 2πu(1−1/ζ), J(−ζ) is the incomplete gamma function Γ(0,−ζ), and ζ(u) is the unique solution of e^ζ=−uζ in a specified region, equivalently the 1-branch of the Lambert W function applied to 1/u. The error is bounded by 420 u^{−6} for 6≤u≤19 and by 0.005 u^{−2} for u≥16; a simpler corollary gives |W(u)|<2|Φ(u)| for u≥3. The same method yields a refinement with error below 0.01 u^{−3} on 18≤u≤1000 and below 8.4 u^{−3} for larger u. This is achieved by the saddle point method: the Laplace transform of ω is e^{J(s)}−1, the contour i

Load-bearing premise

The proof relies on a chain of finite-range inequalities verified by graphing, contour plots, and numerical integration rather than by written analytic proof, and if any one of those unchecked numerical checks is wrong, the corresponding range of the theorem is unsupported.

Editorial extensions

If this is right

For any u≥3, ω(u) can be bracketed rigorously by evaluating a closed-form expression, eliminating the need to solve (uω(u))'=ω(u−1) numerically for a rigorous bound.
The bounds plug directly into the prime-short-interval formulas: the limsup and liminf of normalized prime counts in short intervals are trapped between explicit numbers built from max and min of 2|Φ|cos(arg Φ) plus a known error.
The corollary |W(u)|<2|Φ(u)| supplies a clean, universally valid envelope that may simplify future arguments needing an unconditional bound on ω(u)−e^{−γ}.
The third refinement gives an O(u^{−3}) error uniformly for u≥1000, so high-precision values of ω(u) are obtainable from the formula alone.
Because Φ(u) is expressed through Γ(0,−ζ) and W_1(1/u), standard numerical libraries already provide the ingredients; no delay-equation solver is needed.

Reading between the lines

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

A likely consequence the author leaves implicit: the same saddle-point expansion should extend to the wider family of differential-difference equations of the form uf'(u)+af(u)+bf(u−1)=0, giving explicit bounds whenever the associated saddle point is simple; the complex-versus-real character of the saddle point is the main new ingredient to manage.
The finite-range checks that fill the gap where the analytic estimates do not reach are asserted graphically; replacing those plots with certified interval arithmetic would turn the result into a fully machine-checkable proof and would also fix the sharpest constant across the entire range.
One testable extension: the formulas can be used to compute rigorous confidence intervals for ω(u) at its local extrema, sharpening the published numerical tables beyond u=11 and thereby making the short-interval prime oscillation statement quantitative at larger ranges.
If the bounds are as tight as stated, they imply that W(u) oscillates in sign infinitely often with amplitude decaying very fast, reinforcing the heuristic that the distribution of rough numbers tracks its smooth main term to an exponentially small relative error.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Bottom line: this paper gives the first numerically explicit, easy-to-evaluate two-sided bounds for W(u)=omega(u)-e^{-gamma}, with error terms of order 1/(u log u) and 1/u^2. The main term uses the Lambert W branch and incomplete gamma function, so it is straightforward to compute in Mathematica or PARI/GP. The saddle-point derivation is careful, with Taylor expansions to order seven, explicit lemmas, and no fitted parameters. The correction terms are explicit rational functions of zeta. That is a real step beyond Hildebrand's asymptotic O(1/u) and Cheer-Goldston's numerical extrema.

What the paper does well: the large-u analysis (roughly u >= 10^3 for Theorem 1, u >= 5500 for Theorem 2) is self-contained and convincing. Lemma 4 gives sharp bounds for J^{(k)}; Lemma 5 controls the decay of the Laplace integral away from the saddle point; the Taylor expansion around the saddle is handled carefully. Corollary 1 gives a clean bound |W(u)| < 2|Phi(u)| for all u >= 3. I agree with the reader that there is no circularity — the parameters are not fitted to the target values.

Where it gets shaky: the proof of the theorems for the full stated range depends on a long list of finite-range verifications done "by graphing", by contour plots, or by Mathematica numerical integration — Lemmas 3, 5, 7, the proof of Theorem 2 for 6 <= u <= 5500, and Section 4 for u <= 50 via the Marsaglia-Zaman-Marsaglia DDE solver. None of these checks is accompanied by code, output, or error bounds. If any one of them is wrong or merely imprecise, the corresponding range of the theorems is unsupported. The author is transparent that these are numerical checks, not analytic proofs, but the reader cannot audit them. This is a reproducibility gap, not an identified mathematical error. For a paper whose selling point is "no need to solve the DDE", relying on an unprovided DDE solve for u <= 50 and unprovided Mathematica integration for the intermediate range is a little ironic.

My take: the analytic core is solid and the small-range checks are probably right — the author has a track record with the analogous Dickman paper. But the absence of code and data makes the full-range theorem less than fully verified. For a serious referee, I would ask the author to supply a notebook with the numerical verifications, or better, replace the contour-plot arguments with interval-arithmetic or polynomial-certificate checks.

The paper should go to peer review, but with a request for the verification artifacts. It is a useful, citable tool for sieve theory and Maier-type applications.

Best, [Sign-off]

Referee Report

4 major / 5 minor

Summary. The paper establishes explicit, easily evaluated upper and lower bounds for Buchstab's function W(u)=ω(u)−e^{-γ}, using the saddle-point method with a carefully chosen complex valued saddle point ζ=ζ(u) satisfying e^ζ=-uζ. The main results are Theorems 1–3, which express W(u) as 2|Φ_j(u)|{cos(arg Φ_j(u))+θ_j(u)} with explicit numerical error bounds; Theorem 2 is the most refined two-term approximation, and Theorem 3 adds a further β(u) term. The claimed advantage is that the resulting bounds involve only the incomplete gamma function Γ(0,s) and the Lambert W function, so no numerical solution of the delay-differential equation is required. The analytic saddle-point derivation in §2 is detailed, with Taylor polynomials of order seven, explicit lemmas estimating the derivatives of J(s), and large-u error terms derived. However, the full range of the theorems depends substantially on finite-range numerical verifications described in the text as 'graphing', 'contour plots', and Mathematica numerical integration, without supplying code, data, or rigorous error bounds.

Significance. If fully supported, the paper would be a valuable contribution: it gives simple, explicit two-sided estimates for Buchstab's function that improve on Hildebrand's asymptotic formula and avoid numerical DDE solving. The main-term formulas are parameter-free, use standard special functions, and are easy to evaluate in Mathematica or PARI/GP. The explicit constants, e.g. |θ_2(u)|<0.005u^{-2} for u≥16, are strong enough for applications such as Maier-type prime gap estimates. The analytic core appears sound and is a nontrivial extension of the author's earlier Dickman-function work to the complex saddle-point setting. The main weakness is not the analytic derivation for large u, but the reproducibility and verifiability of the finite-range checks that close the gap between the large-u proof and the claimed full range u≥3.

major comments (4)

[§4, proof of Theorem 2] The proof of Theorem 2 for 6≤u≤5500 relies entirely on numerical computation: the Marsaglia–Zaman–Marsaglia DDE solver for u≤50 and Mathematica numerical integration of S(u) for 50≤u≤5500. The analytic bound |θ_2(u)|<0.005u^{-2} is derived only for u≥5500, so the theorem as stated for all u≥16 (and particularly the 16≤u≤5500 range) is supported only by these unaudited computations. Please supply the actual code, output data, and rigorous error bounds (e.g. interval arithmetic or explicit quadrature error estimates) for these verifications. Without that, the full-range claim is not reproducible.
[Lemma 3] The proof of (2) in the rectangle 0≤x≤200, 3≤y≤400 is asserted to follow from a contour plot, and this is used to establish the bound Re(J(−ζ))≥u for u≤80. Since this finite-range check is load-bearing for the small-u behavior of the later estimates, the contour plot should be replaced by a machine-checkable verification, or by a rigorous explicit bound valid in that rectangle.
[Lemma 5] The inequalities Re(H(u,τ))/u≤−0.39 for 50≤u≤10^3, 1≤|τ|≤4, and Re(H(u,τ))<−8u/log^2 u for 50≤u≤10^4, 4≤τ≤370, are verified 'by graphing'. These estimates are not peripheral: they control the entire contribution of the integration ranges |τ|≥δ in the saddle-point argument and are used to derive (9) for u≥10^3. Please supply a rigorous or reproducible verification (code, data, or exact bounds) for these rectangles.
[Lemma 7 and Theorem 1/Corollary 1] The bound |f(ζ)|<1 is verified by a contour plot for 5≤x≤10, and the inequality in Lemma 7 is checked by graphing for 2≤u≤25. This lemma is used in the proof of Theorem 1 and in the estimate |1+α(u)|>1−1/(10u), which is needed for the coefficient 0.005 in Theorem 2. Similarly, Corollary 1's verification for 3≤u≤6 is done by graphing a quotient. These finite-range verifications should be made explicit and machine-checkable, since they are part of the logical proof of the stated theorems.

minor comments (5)

[§1, Table 1] The notation 'a: Thm. 1' and 'a: Thm. 2' is a little cryptic; please clarify that the tabulated entries are a×10^{-b} with a truncated/rounded as stated.
[§2.2] The quantity f(u) is introduced as u^3(E_2(u)+∑_{k=1}^6 E_k(u)), but the definition of E_k(u) for k=1,...,6 is spread over the proof; a short summary equation would improve readability.
[Theorem 3] The ranges '18≤u≤10^3' and 'u≥10^3' overlap at u=10^3; since the two bounds differ, please clarify which bound is intended at u=10^3.
[General] Figure 1 is referenced in the text but no actual figure appears in the manuscript. Please include the plot or remove the reference.
[§4] The statement 'The proof of Theorem 2 shows that the last integrand approaches the standard normal density function' is informal; since the paper is about rigorous bounds, a precise quantified statement would be preferable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the saddle-point expansion is derived from external Laplace-transform theory, and the finite-range checks are independent numerical verifications, not fitted or self-referential inputs.

full rationale

The derivation chain is self-contained in the relevant sense: define ζ by e^ζ = -uζ, define J(s)=Γ(0,s) and Φ(u) from it, use Tenenbaum's Laplace-transform representation ω̂(s)=e^{J(s)}-1, move the contour, expand J(-ζ+iτ) near the saddle point τ=0, bound the tails with Lemmas 3-5, and assemble W_+(u)=Φ(u)(1+α(u)+β(u)+λ(u)u^{-3}). The main-term coefficients α and β are computed explicitly from derivatives of J, not fitted to values of W(u); no parameter is adjusted to make the theorem match the target function. The small-u and intermediate-u ranges are checked by independent numerical computation: the Marsaglia-Zaman-Marsaglia DDE solver for u≤50, and Mathematica numerical integration of the inverse-Laplace integral representation for 50≤u≤5500. These checks compute W(u) separately and compare it with the proposed Φ_i formulas, so they could in principle falsify the claims; they are not manufactured from the claims. The only self-citation is [10, Lemma 1], an elementary bound on I^(k)(µ)=∫_0^1 h^{k-1}e^{hµ}dh from the author's Dickman-function paper, used to control |J^(k)(-ζ+iτ)| in Lemma 4. That is an auxiliary estimate about a real integral, not the Buchstab target, and it does not assume the present theorems; it is independent supporting content rather than a circular premise. Therefore no circular step of any of the enumerated kinds is present. The 'by graphing', contour-plot, and Mathematica-integration verifications are not accompanied by code or data, which is a reproducibility gap and a correctness-risk concern, but not a reduction of a prediction to an input within the paper's own equations.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a known asymptotic decomposition plus two auxiliary estimates from the literature and, crucially, on a collection of unverifiable finite-range numerical checks. There are no new physical or mathematical entities and no target-fitted parameters; the main contribution is packaging an existing saddle-point expansion into explicit bounds.

free parameters (1)

Explicit error constants in Theorems 1–3
The quantitative bounds (e.g., 1/(12u log u), 420u^{-6}, 0.005u^{-2}, 8.4u^{-3}) are hand-chosen envelope constants whose small-range validity is confirmed numerically rather than derived. They are not physical fitted parameters, but the theorem's content as stated depends on them.

assumptions (5)

domain assumption Hildebrand's saddle-point representation W(u)=2|Φ(u)|{cos(arg Φ(u))+O(1/u)}
Central starting expansion, cited to [3]; the paper sharpens the main term and error but does not independently reprove the base representation.
domain assumption Laplace transform identity bω(s)=e^{J(s)}-1 with only pole at 0 and residue e^{-γ}
Invoked via Tenenbaum [9, Thm. III.6.7] to set up the inverse Laplace integral in §2.2.
standard math Pinelis upper bound for the incomplete gamma function, as used in Lemma 6
Published analytic estimate [7, Prop. 2.7], used to bound tails of Gaussian integrals.
domain assumption Weingartner [10, Lemma 1] bound on I^{(k)}(μ)
Self-cited estimate for Dickman's function used in Lemma 4; it is independent of the present target, so not circular.
ad hoc to paper Finite-range inequalities verified by graphing/contour plots and Mathematica are correct
Locations: Lemma 3 (rectangle 0≤x≤200, 3≤y≤400; u≤80), Lemma 5 (50≤u≤10^4, 4≤τ≤370; 50≤u≤10^3), Lemma 7 (2≤u≤25; 5≤x≤10), proof of Theorem 2 (860≤u≤10^6), Section 4 (u≤5500). No code or data supplied.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Explicit bounds for Buchstab's function." pith.science (2026). https://pith.science/paper/AMECFK2B

@misc{pith2026260721883,
  author       = {Pith},
  title        = {Pith review of: Explicit bounds for Buchstab's function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMECFK2B}},
  note         = {Machine review of arXiv:2607.21883}
}

abstract

Buchstab's function $\omega(u)$ describes the distribution of integers without small prime factors. We establish numerically explicit upper and lower bounds for $\omega(u)$ that are easy to evaluate, without the need to solve the delay differential equation numerically.

Figures

Figures reproduced from arXiv: 2607.21883 by the authors.

**Figure 1.** shows the graphs of θ1(u) from Theorem 1, θ2(u) from Theorem 2, and the bounds ± 1 12u log u from Theorem 1, for 6 < u ≤ 12. Theorem 3 below exhibits an even smaller error term than Theorem 2, at the expense of a more complicated main term. 7 8 9 10 11 12 u -0.005 0.005 θ1(u) θ2(u) 1 12 u log(u) - 1 12 u log(u) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 1 linked inside Pith

[1]

A. A. Buchstab, Asymptotic estimates of a general number-theoretic function,Mat. Sb.44(1937), 1239–1246

1937
[2]

A. Y. Cheer and D. A. Goldston, A differential delay equation arising from the sieve of Eratosthenes,Math. Comp.55(1990), no. 191, 129–141

1990
[3]

Hildebrand, The asymptotic behavior of the solutions of a class of differential- difference equations,J

A. Hildebrand, The asymptotic behavior of the solutions of a class of differential- difference equations,J. London Math. Soc. (2)42(1990), no. 1, 11–31

1990
[4]

Hildebrand and G

A. Hildebrand and G. Tenenbaum, On a class of differential-difference equations arising in number theory,J. Anal. Math.61(1993), 145–179

1993
[5]

Maier, Primes in short intervals,Michigan Math

H. Maier, Primes in short intervals,Michigan Math. J.32(1985), no. 2, 221–225. EXPLICIT BOUNDS FOR BUCHSTAB’S FUNCTION 15

1985
[6]

Marsaglia, A

G. Marsaglia, A. Zaman and J. C. W. Marsaglia, Numerical solution of some classical differential-difference equations.Math. Comp.53(1989), no. 187, 191–201

1989
[7]

Pinelis, Exact lower and upper bounds on the incomplete gamma function,Math

I. Pinelis, Exact lower and upper bounds on the incomplete gamma function,Math. Inequal. Appl.23(2020), no. 4, 1261–1278

2020
[8]

Tenenbaum, Crible d’ ´Eratosth` ene et mod` ele de Kubilius,Number theory in progress, Vol

G. Tenenbaum, Crible d’ ´Eratosth` ene et mod` ele de Kubilius,Number theory in progress, Vol. 2 (Zakopane-Ko´ scielisko, 1997), 1099–1129, de Gruyter, Berlin, 1999

1997

Show all 10 references

[9]

Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, Graduate Studies in Mathematics, Vol

G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, Graduate Studies in Mathematics, Vol. 163, American Mathematical Society, 2015

2015
[10]

A. Weingartner, Explicit bounds for Dickman’s function, arXiv:2606.07785 Department of Mathematics, Southern Utah University, 351 West Univer- sity Boulevard, Cedar City, Utah 84720, USA Email address:weingartner@suu.edu

arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

A. A. Buchstab, Asymptotic estimates of a general number-theoretic function,Mat. Sb.44(1937), 1239–1246

1937

[2] [2]

A. Y. Cheer and D. A. Goldston, A differential delay equation arising from the sieve of Eratosthenes,Math. Comp.55(1990), no. 191, 129–141

1990

[3] [3]

Hildebrand, The asymptotic behavior of the solutions of a class of differential- difference equations,J

A. Hildebrand, The asymptotic behavior of the solutions of a class of differential- difference equations,J. London Math. Soc. (2)42(1990), no. 1, 11–31

1990

[4] [4]

Hildebrand and G

A. Hildebrand and G. Tenenbaum, On a class of differential-difference equations arising in number theory,J. Anal. Math.61(1993), 145–179

1993

[5] [5]

Maier, Primes in short intervals,Michigan Math

H. Maier, Primes in short intervals,Michigan Math. J.32(1985), no. 2, 221–225. EXPLICIT BOUNDS FOR BUCHSTAB’S FUNCTION 15

1985

[6] [6]

Marsaglia, A

G. Marsaglia, A. Zaman and J. C. W. Marsaglia, Numerical solution of some classical differential-difference equations.Math. Comp.53(1989), no. 187, 191–201

1989

[7] [7]

Pinelis, Exact lower and upper bounds on the incomplete gamma function,Math

I. Pinelis, Exact lower and upper bounds on the incomplete gamma function,Math. Inequal. Appl.23(2020), no. 4, 1261–1278

2020

[8] [8]

Tenenbaum, Crible d’ ´Eratosth` ene et mod` ele de Kubilius,Number theory in progress, Vol

G. Tenenbaum, Crible d’ ´Eratosth` ene et mod` ele de Kubilius,Number theory in progress, Vol. 2 (Zakopane-Ko´ scielisko, 1997), 1099–1129, de Gruyter, Berlin, 1999

1997