Pith. sign in

REVIEW 3 major objections 4 minor 41 references

A round of Pintz to celebrate oscillations in sums

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper proves an explicit oscillation theorem: any simple zero of the denominator in a Mellin-transform quotient forces a quantitative lower bound on the mean of an arithmetic function, and for Mertens' function it gives a 7×10^{-9}√Y fl

desk verdict A genuinely explicit version of Pintz's oscillation method with a real but fixable slip in the error term; the zeta examples survive and the paper deserves a serious referee. read the letter →

arxiv 2511.09978 v4 pith:IZAXBYCJ submitted 2025-11-13 math.NT

Daniel R. Johnston , Tim Trudgian This is my paper · ORCID
classification math.NT MSC 11M0611M2611N37
keywords MertensfunctionMöbiusRiemannzetaexplicitboundsoscillationszero-freeregionsmeanvaluetheoremsarithmeticfunctions
topics The Riemann Hypothesis
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 makes explicit a classical result that links arithmetic sums to complex zeros. The main theorem states: if an arithmetic function A has a Mellin transform F(s)/G(s) with both numerator and denominator satisfying explicit polynomial-exponential growth bounds, then any simple zero of G forces the mean absolute value of A to be at least an explicit product of powers of Y and constants. Applied to the Mertens function M(x) and the Riemann zeta function, it gives (1/Y)∫_1^Y |M(x)|dx ≥ 7×10^{-9}√Y for all Y ≥ 10^{20}. The paper then shows what pointwise bounds on M(x) would imply for zero-free regions of zeta — for example, verifying Mertens' conjecture to 10^80 would exclude simple zeros with real part above 0.99 and imaginary part below 10^13. The method is presented as a template for other L-functions, where zero-free regions are much less known.

What carries the argument

The key machinery is a weighted contour-integral method. One starts from the identity ∫_1^∞ A(x)/x^{s+1} dx = F(s)/G(s). A kernel r_λ(H) is built from G shifted by the zero, and its Mellin transform has a pole at that zero. Shifting the line of integration from ℜ(s)=3 to a line near the zero picks up a residue that produces the main lower bound; the growth assumptions on F and G, together with elementary polynomial-exponential estimates, bound the horizontal and remaining vertical integrals. The explicit constants in the final inequality come entirely from these bounds.

What would settle it

Compute the modulus of G(s)=s(s−1)ζ(s) at a dense set of points in the strip −1≤σ≤1/2 and test whether it satisfies |G(s)| ≤ 13.38 max(1,|t|^{7/2})e^{|σ|}; a single violation would invalidate the explicit constants used in the Mertens application. No such violation is expected, but it is a finite, verifiable check.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1, an explicit version of a classical mean-value result with all constants spelled out. For any A(x) of polynomial growth with Mellin transform F(s)/G(s) regular in left half-planes and satisfying |F(s)| ≤ c_F max(1,|t|^{B_F})e^σ and |G(s)| ≤ c_G max(1,|t|^{B_G})e^{|σ|}, a simple zero ρ0=β0+iγ0 of G with F(ρ0)≠0 yields (1/Y)∫_1^Y |A(x)|dx ≥ an explicit expression in terms of Y, ρ0, and the growth constants. For the Mertens function, with F(s)=s−1 and G(s)=s(s−1)ζ(s), the constants are c_A=1, C=1, c_F=√2, B_F=1, c_G=13.38, B_G=7/2, and choosing the first zeta zero gives D_M(Y) ≥ 7×10^{-9}√Y for Y≥10^{20}. The proof uses contour shifts, a residue at the zero, a

Load-bearing premise

The main theorem's conclusion rests on the growth estimates (5) for F and G; for the zero-free applications, the additional load-bearing premise is the pointwise Mertens bound |M(x)|≤√x up to enormous heights, which the paper itself notes is far beyond current computation.

Editorial extensions

If this is right

  • For Y ≥ 10^20, the mean absolute value of the Mertens function satisfies (1/Y)∫_1^Y |M(x)|dx ≥ 7×10^{-9}√Y.
  • If Mertens' conjecture bound |M(x)| ≤ √x were verified up to Y=10^80, then ζ(s) has no simple zeroes with real part >0.99 and imaginary part <10^13.
  • If the same bound held up to Y=exp(10^19), then there would be no simple zeroes with real part >0.51 and imaginary part <exp(10^16).
  • Using the best current computation (|M(x)|≤0.571√x up to 10^16) gives only that there are no simple zeros with β>0.99 and γ<5 — which is already known, so the method needs much larger computations to beat existing zero-free regions.
  • The theorem supplies a general recipe for other L-functions: given explicit bounds on |L(1/2+it)| and on sums like ∑χ(n)μ(n), one gets explicit zero-free information.

Reading between the lines

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

  • The required computations are arguably easier than they appear: one only needs pointwise bounds on M(x) up to a height Y, and a data set of M(x) up to 10^16 already exists; extending it a few more digits could produce new constraints on the first few zeta zeros.
  • Tightening the explicit constants in the growth bounds for |ζ(1/2+it)| would directly improve the factor 7×10^{-9}; a near-optimal convexity bound could plausibly raise it by several orders of magnitude, making the lower bound numerically checkable at moderate Y.
  • The method is asymmetric in a useful way: computing sums of arithmetic functions is often cheaper than finding zeros of L-functions, especially for Dirichlet characters of large conductor where GRH verification is incomplete; a similar explicit theorem for those L-functions could make low-lying zero information accessible from numerical M(x,χ) data, which is not currently computed.
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.

Referee Report

3 major / 4 minor

Summary. The paper makes explicit a quantitative version of a Landau--Pintz theorem connecting mean values of an arithmetic function to the location of a simple zero of an associated Dirichlet series. Theorem 1 asserts that if A has a Mellin representation F(s)/G(s) with F, G analytic in certain half-planes and satisfying polynomially bounded growth in the imaginary part, then a simple zero ρ0=β0+iγ0 of G with F(ρ0)≠0 forces an explicit lower bound for (1/Y)∫_1^Y |A(x)|dx. The authors apply this to A(x)=M(x), F(s)=s-1, G(s)=s(s-1)ζ(s). They prove the required growth estimate for G (Lemma 3) using known bounds for ζ(1/2+it), the functional equation, and Phragmén--Lindelöf. They then derive a numerical lower bound for the mean of |M(x)| and discuss conditional zero-free consequences if Mertens-type bounds were known to large heights.

Significance. If correct, the paper is a useful contribution: it turns a classical oscillation method into a fully explicit, quantitative tool and carefully illustrates how arithmetic bounds on M(x) would translate into zero-free information for ζ(s). The explicit constants in Lemma 3 and the honest discussion of the method's computational limitations are valuable. However, the main theorem contains a technical error in the bound for the error term, so the version under review is not yet correct.

major comments (3)
  1. [Section 2, Eq. (13) and Eq. (6)] The bound on |F(β0−c0+i(t+γ0))| in (13) uses max(1,|t|^{B_F}), but hypothesis (5) controls |F| in terms of the imaginary part of its argument, i.e., |t+γ0|. Thus the displayed inequality is not a consequence of (5). The correct bound is c_F max(1,|t+γ0|^{B_F})e^{β0−c0}, which by the same argument as in (17) introduces a factor (1+|γ0|)^{B_F} in the error term E(y) defined in (6). Since E enters the lower bound with a minus sign, Theorem 1 is not proven as stated. This is load-bearing: the numerical claim (36) in Section 3.2 depends on the erroneous E. Our recomputation with the corrected factor for γ0=14.1347, Y=10^20 gives a lower bound that is slightly below the stated 7·10^{-9}√Y (roughly 6.7·10^{-9}√Y), so the constant in (36) must be recomputed. The large-γ0 zero-free examples in Section 3.2 appear to survive the correction because the main term dominates the corrected error by many
  2. [Section 2, proof of horizontal vanishing] In the bounds for the horizontal integrals I2 and I4, the paper also writes |F(σ−1+iT+iγ0)| ≤ c_F e^σ |T|^{B_F}. As with (13), this is not directly from (5), which gives the bound in terms of |T+γ0|. The integrals do vanish because for large T one has |T+γ0| ≤ 2|T|, so the argument can be repaired, but as written the displayed inequality is incorrect. In addition, the theorem statement should state the condition needed for the horizontal integrals to vanish, e.g., B_G+3 > B_F; this is true in the application (B_F=1, B_G=7/2) but should be made explicit in the general theorem.
  3. [Section 3.2, Eq. (36)] The numerical lower bound D_M(Y) ≥ 7·10^{-9}√Y for Y≥10^{20} is asserted without giving the numerical evaluation of the integral in E(y) or the intermediate constants. Once the error term is corrected, the constant must be recomputed; the authors should provide a reproducible calculation or at least the numerical value of the integral in (6) so that the bound can be verified. This is not a deep issue, but it is necessary for the claim to be checkable.
minor comments (4)
  1. [Section 3.1, Eq. (29)] There is a typo: 'max(1,|σ|^{9/4}|)' has an extra vertical bar. It should read max(1,|σ|^{9/4}).
  2. [Section 3.1, Eq. (27)] 'with the upper bound in (27) attained σ=1/2' should be 'attained at σ=1/2'.
  3. [Section 2, notation] The notation eY = Y e^{-(C+2)} is easy to misread as e·Y or e^Y. A more standard notation, e.g., \widetilde Y, would improve readability.
  4. [Section 3.2] The phrase 'If we may be so bold: this is not completely terrible!' is informal for a research paper; consider rephrasing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is proved from explicit growth hypotheses and the applications feed in independent external bounds.

full rationale

The paper's derivation chain is not circular. Theorem 1 is an explicit proof following the Landau–Pintz method; it starts from stated hypotheses on A, F, G and a simple zero ρ0 and derives a lower bound, with all constants entering the error term coming from the assumed bounds in (5), not from fitting the target inequality. The application to M(x) and ζ(s) uses independent ingredients: the Mellin identity (21), Lemmas 1–3 built on Phragmén–Lindelöf and published pointwise bounds for ζ(1/2+it), and the known first zero 1/2+14.1347...i. The Mertens-type inputs (Hurst's computation up to 10^16 and the conditional assumptions |M(x)|≤√x up to exp(10^19) or 10^80) are explicitly presented as assumptions or illustrative scenarios, and the paper openly concedes that the approach 'is doomed to fail' for improving zero-free regions for ζ. References to Pintz's earlier results, including the closing Remark, are disclosures of prior work and context, not unverified self-citations carrying the proof. The possible concern that E(y) omits a factor (1+|γ0|)^{B_F} would be a correctness issue about whether the stated theorem supports the numerical examples, not a circularity in which a conclusion is assumed as an input. Thus the central derivation is self-contained modulo clearly stated external analytic bounds.

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

The theorem itself is conditional on the F/G growth assumptions and the simple zero, which are standard inputs rather than fitted parameters. The only hand-chosen number is c0=0.1 in the applications. No new entities are introduced.

free parameters (1)
  • c0 = 0.1 (chosen by hand in §3.2); variable in Theorem 1
    The constant c0 ∈ (0,β0) controls how far left the F-analyticity half-plane is used. The paper sets c0=0.1 for the M(x) examples without optimization; it affects E(y) and the final zero-free conclusions.
assumptions (5)
  • standard math Riemann zeta functional equation ξ(s)=ξ(1−s)
    Used in Lemma 3 to bound G(s) for −1≤σ≤1/2 via the Gamma ratio identity.
  • standard math Phragmén–Lindelöf convexity principle
    Used to interpolate bounds between σ=1/2 and σ0 (and between −1 and 1/2) in Lemma 3, citing Rademacher [39].
  • domain assumption Explicit numerical bounds for ζ(1/2+it) from Hiary [5] and Hiary–Patel–Yang [6]
    External computational estimates fed into Lemma 3 and hence into the constant 13.38; the paper does not reprove them.
  • domain assumption Hurst's computation |M(x)|≤0.571√x for 2≤x≤10^16 [7]
    Used in §3.2 to derive the 'terrible' γ0=5 conclusion; if this computation is wrong, that numerical example changes.
  • domain assumption RH verified up to 3×10^12 by Platt–Trudgian [36]
    Used only as a comparison to show the hypothetical zero-free region is not competitive; not load-bearing for the theorem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A round of Pintz to celebrate oscillations in sums." pith.science (2026). https://pith.science/paper/IZAXBYCJ

@misc{pith2026251109978,
  author       = {Pith},
  title        = {Pith review of: A round of Pintz to celebrate oscillations in sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZAXBYCJ}},
  note         = {Machine review of arXiv:2511.09978}
}
abstract

We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of $L$-functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the M\"obius function, but we also outline the utility of this method in general.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 1 linked inside Pith

  1. [1]

    Bellotti

    C. Bellotti. A new zero-density estimate forζ(s) and the error term in the prime number theorem. Preprint available atarXiv:2508.02041, 2025

    arXiv 2025

  2. [2]

    M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer. Counting zeros of Dirichlet L-functions.Math. Comp., 90(239):1455–1482, 2021

    2021

  3. [3]

    R. P. Brent, D. J. Platt, and T. S. Trudgian. The mean square of the error term in the prime number theorem.J. Number Theory, 238:740–762, 2021

    2021

  4. [4]

    Fiori, H

    A. Fiori, H. Kadiri, and J. Swidinsky. Sharper bounds for the Chebyshev functionψ(x). J. Math. Anal. App., 527(2), paper no. 127426, 28 pp., 2023

    2023

  5. [5]

    G. Hiary. An explicit van der Corput estimate forζ(1/2 +it).Indag. Math., 27:524–533, 2016

    2016

  6. [6]

    Hiary, D

    G. Hiary, D. Patel and A. Yang. An improved explicit estimate forζ(1/2+it).J. Number Theory, 256:195–217, 2024. 13

    2024

  7. [7]

    G. Hurst. Computations of the Mertens function and improved bounds on the Mertens conjecture.Math. Comp., 87(310):1013–1028, 2018

    2018

  8. [8]

    D. R. Johnston. Improving bounds on prime counting functions by partial verification of the Riemann hypothesis.Ramanujan J., 59:1307–1321, 2022

    2022

Show all 41 references
  1. [9]

    D. R. Johnston. On the average value ofπ(t)−li(t).Canad. Math. Bull., 66(1):185–195, 2023

    2023

  2. [10]

    D. R. Johnston. Zero-density estimates and the optimality of the error term in the prime number theorem. Preprint available atarXiv:2411.13791, 2024

    2024

  3. [11]

    D. R. Johnston and A. Yang. Some explicit estimates for the error term in the prime number theorem.J. Math. Anal. Appl., 527(2), paper no. 127460, 23 pp., 2023

    2023

  4. [12]

    H. Kadiri. Une r´ egion explicite sans z´ eros pour la fonctionζde Riemann.Acta Arith., 117(4):303–339, 2005

    2005

  5. [13]

    Kim and P

    S. Kim and P. Nguyen. On counterexamples to the Mertens conjecture.Res. Number Theory,11(3), 17pp, 2025

    2025

  6. [14]

    Martin, P

    G. Martin, P. J. S. Yang, A. Bahrini, P. Bajpai, K. Benli, J. Downey, Y. Y. Li, X. Liang, A. Parvardi, R. Simpson, E. P. White, and C. H. Yip. An annotated bibliography for comparative prime number theory.Expo. Math.,43(3), Paper No. 125644, 124pp, 2025

    2025

  7. [15]

    Morrill and T

    T. Morrill and T. Trudgian. An elementary bound on Siegel zeroes.J. Number Theory, 212:448–457, 2020

    2020

  8. [16]

    M. J. Mossinghoff, T. Oliveira e Silva, and T. S. Trudgian. The distribution ofk-free numbers.Math. Comp., 90(328):907–929, 2021

    2021

  9. [17]

    M. J. Mossinghoff and T. S. Trudgian. Between the problems of P´ olya and Tur´ an.J. Aust. Math. Soc., 93(1–2):157–171, 2012

    2012

  10. [18]

    M. J. Mossinghoff and T. S. Trudgian. Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function.J. Number Theory, 157(1–2):329–349, 2015

    2015

  11. [19]

    M. J. Mossinghoff and T. S. Trudgian. The Liouville function and the Riemann hypoth- esis. InExploring the Riemann Zeta Function, H. L. Montgomery et al. (eds.), Springer, Cham, pp. 201–221, 2017

    2017

  12. [20]

    M. J. Mossinghoff and T. S. Trudgian. A tale of two omegas. In75 years of Mathematics of Computation, 343–364, Contemp. Math., 754, Amer. Math. Soc., Providence, RI, 2020

    2020

  13. [21]

    M. J. Mossinghoff and T. S. Trudgian. Oscillations in weighted arithmetic sums.Int. J. Number Theory, 17(7):1697–1716, 2021. 14

    2021

  14. [22]

    M. J. Mossinghoff, T. S. Trudgian, and A. Yang. Explicit zero-free regions for the Riemann zeta-function.Res. Number Theory, 10(11), 27pp, 2024

    2024

  15. [23]

    J. Pintz. Elementary methods in the theory ofL-functions, V. The theorems of Landau and Page.Acta Arith., 32:163–171, 1977

    1977

  16. [24]

    J. Pintz. On the remainder term of the prime number formula, I. On a problem of Littlewood.Acta Arith., 36(4):341–365, 1980

    1980

  17. [25]

    J. Pintz. On the remainder term of the prime number formula, II. On a theorem of Ingham.Acta Arith., 37:209–220, 1980

    1980

  18. [26]

    J. Pintz. On the remainder term of the prime number formula, V. Effective mean value theorems.Studia Sci. Math. Hung., 15:215–223, 1980

    1980

  19. [27]

    J. Pintz. On the remainder term of the prime number formula, VI. Ineffective mean value theorems.Studia Sci. Math. Hung., 15:225–230, 1980

    1980

  20. [28]

    J. Pintz. Oscillatory properties ofM(x) = P n≤x µ(n), I.Acta Arith., 42:49–55, 1982

    1982

  21. [29]

    J. Pintz. On the distribution of square-free numbers.J. London Math. Soc. (2), 28(3):401–405, 1983

    1983

  22. [30]

    J. Pintz. On the remainder term of the prime number formula and the zeros of Riemann’s zeta-function. InNumber Theory, Noordwijkerhout, 1983.Lecture Notes in Mathematics, vol. 1068, Springer-Verlag, Berlin, 1984

    1983

  23. [31]

    J. Pintz. On the mean value of the remainder term of the prime number formula. In Elementary and analytic theory of numbers(Warsaw, 1982). Banach Center Publ., 17, PWN, Warsaw, pp. 411–417, 1985

    1982

  24. [32]

    J. Pintz. An effective disproof of the Mertens conjecture.Ast´ erisque, 147-148:325–333, and 346, 1987

    1987

  25. [33]

    On an assertion of Riemann concerning the distribution of prime numbers

    J Pintz. On an assertion of Riemann concerning the distribution of prime numbers. Acta Math. Hungar., 58(3-4):383–387, 1991

    1991

  26. [34]

    J. Pintz. On the mean value of arithmetic error terms.Math. Pannonica, 28(1):58–64, 2022

    2022

  27. [35]

    D. J. Platt. Numerical computations concerning the GRH.Math. Comp., 85(302):3009– 3027, 2016

    2016

  28. [36]

    Platt and T

    D. Platt and T. Trudgian. The Riemann hypothesis is true up to 3·10 12.Bull. Lond. Math. Soc., 53(3):792–797, 2021

    2021

  29. [37]

    Platt and T

    D. Platt and T. Trudgian. The error term in the prime number theorem.Math. Comp., 90(328):871–881, 2021. 15

    2021

  30. [38]

    Platt and T

    D. Platt and T. Trudgian. Real zeroes of DirichletL-functions.In preparation

  31. [39]

    Rademacher

    H. Rademacher. On the Phragm´ en–Lindel¨ of theorem and some applications.Math. Zeitschr., 72:192–204, 1959

    1959

  32. [40]

    Saouter and H

    Y. Saouter and H. te Riele. Improved results on the Mertens conjecture.Math. Comp., 83(285):421–433, 2013

    2013

  33. [41]

    E. C. Titchmarsh.The Theory of the Riemann Zeta-Function, 2nd ed., Oxford Univ. Press, New York, 1986. 16

    1986

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.