On the mean values of the error terms in Mertens' theorems

Tianyu Zhao

REVIEW 2 major objections 5 minor 28 references

On the mean values of the error terms in Mertens' theorems

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Riemann hypothesis is equivalent to a positivity condition on Mertens error terms.

desk verdict Theorem 1 is a solid new equivalence between RH and positivity of mean Mertens error terms; the Theta=1 case for E3 rests on a sketched lemma that needs a full proof. read the letter →

arxiv 2411.18903 v2 pith:SGTBCBNF submitted 2024-11-28 math.NT

Tianyu Zhao This is my paper · ORCID

classification math.NT MSC 11M0611M2611N0511Y35

keywords Mertens'theoremsRiemannhypothesisGeneralizederrortermsexplicitformulaLandau'soscillationtheoremDirichletcharactersarithmeticprogressions

topics The Riemann Hypothesis

open problems 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

The paper claims that the Riemann hypothesis is equivalent, for the first two error terms in Mertens' theorems, to a clean positivity condition: the integral of the error term from 2 to any X must be positive. If true, the most famous conjecture in number theory becomes a statement about the average behavior of prime sums, and the same equivalence extends to the third Mertens error term under an additional zero-free-region regularity assumption. The paper also proves analogues for prime sums twisted by real Dirichlet characters and for primes in arithmetic progressions, and it identifies the complete finite sets of characters and moduli for which the equivalence survives. A sympathetic reader would care because it gives a new formulation of RH that makes no reference to zeros and a template for converting zero-location conjectures into sign conditions on averages.

What carries the argument

The argument is carried by the explicit formula for ψ(x) and the related formula for π(x)-li(x), which turn the integrated error terms into sums over nontrivial zeros of ζ(s). The necessity direction uses Landau's oscillation theorem: if a zero lies off the critical line, the zero sum forces the integral to take both signs. The sufficiency direction is quantitative: under RH, the total zero contribution is controlled by the constant B1=-2 ξ'/ξ(0)=0.0461..., and positivity follows because the dominant term exceeds all remainder terms. For the third Mertens error term, E3 is expanded as an exponential in E2, so the sign of its integral is decided by a race between the first moment of E2, positive by Theorem 1 under RH, and its second moment, which is bounded using a mean-square estimate of ψ(x)-x; in the Θ=1 case a lemma following [17] produces the required large oscillations.

What would settle it

To test Theorem 1, compute the integral from 2 to X of E_1 or E_2 at very large X using the explicit formula under the assumption of RH; a single X with a non-positive value would refute the sufficiency direction. For the Θ=1 case of Theorem 2, one could construct or prove the existence of a zero-free region not representable by any function satisfying Assumption 1, which would block the required control of the second moment.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: for i=1,2, the Riemann hypothesis holds if and only if the integral from 2 to X of the ith Mertens error term is positive for every X>2. For the third Mertens error term, RH implies the same positivity, and if the supremum of the real parts of the nontrivial zeros satisfies 1/2<Θ<1, or Θ=1 under Assumption 1, then the integral changes sign infinitely often. The proof also yields precise asymptotics: with the normalization f_i(X) in (1), the limit inferior and limit superior of f_i(X) are 2-B1 and 2+B1, where B1=-2 ξ'/ξ(0)=0.0461..., so the positivity holds with a fixed numerical gap. For real primitive characters, GRH for L(s,χ_d) is equivalent to the positivity condition for exactly the 178 fundamental discriminants listed in Tables 2 and 3; for arithmetic progressions, GRH modulo q is equivalent to the positivity condition for a=1 exactly for the 24 moduli in Q.

Load-bearing premise

The load-bearing premise is Assumption 1, that the sharp boundary of the zero-free region is a differentiable decreasing function whose derivative grows slowly, which is needed only for the Θ=1 case of Theorem 2; if no such function exists, that particular conclusion is unproved, though Theorem 1 is independent of it.

Editorial extensions

If this is right

If Theorem 1 is correct, RH is equivalent to a positivity statement that can be stated without mentioning zeros: every finite-interval mean of the first two Mertens error terms is positive.
Under RH, Corollary 1 gives a localized version: for any c < (2-B1)/(2+B1))^2 = 0.9548..., the integral from cX to X of each E_i is positive for all large X, so positivity persists on short intervals near X.
If RH fails with a zero of real part Θ>1/2, the integral of E3 oscillates in sign infinitely often, making the positivity phenomenon exactly a zero-location phenomenon.
For real primitive characters, the equivalence holds precisely for the 178 discriminants in D; outside D, assuming linear independence of zero ordinates, GRH implies that positivity eventually fails, so the list is a sharp arithmetic classification.
For arithmetic progressions modulo q with residue 1, the same sharp classification holds for the 24 moduli in Q; for other q, GRH plus linear independence forces the positivity to fail.

Reading between the lines

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

This suggests a general heuristic: for many prime-counting error terms, a zero-location conjecture is equivalent to the positivity of the integrated error, and the dividing line is whether the zero-contribution constant stays below 2; testing this for other arithmetic functions, such as those from number fields, is a natural extension.
The fixed numerical gap 2-B1 = 1.9539... is wide enough that the positivity at moderate X might be verifiable computationally before reaching the asymptotic regime, even though the proof only guarantees it for all X under RH.
The Θ=1 case shows the difficulty is a moment problem: controlling the second moment of E2 via zero-free regions is as hard as controlling the first moment, so any future proof removing Assumption 1 would likely require a new second-moment estimate.
The finite sets D and Q can be viewed as an average-bias classification, and similar finite lists may exist for other L-functions whose analogue of Bχ can be computed explicitly.

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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

Main thing to know: Theorem 1 is a real result and the proof is in good shape. The equivalence of RH with positivity of the mean values of E1 and E2 is new and genuinely extends Johnston's pi-li theorem. I read through the sufficiency and necessity arguments and I don't see a gap. The paper deserves a serious referee. What is actually new: the E1/E2 equivalence, the conditional analysis for E3, and the character and arithmetic progression classifications. The computations of D and Q are backed by Sage code and use explicit zero-counting bounds; that part looks honest and reproducible. The asymptotic formula (2) with the constant B1 is a nice addition, and the localized positivity corollary follows cleanly. Soft spots: Theorem 2 for Theta=1 rests on Assumption 1 and on Lemma 10, which is only sketched as an adaptation of Pintz. The key amplitude in (28) has gamma^{-2-epsilon}; Pintz had gamma^{-1-epsilon} for pi-li. That extra factor is exactly what makes Delta1 dominate Delta2. Since the proof of Lemma 10 is a sketch and the double-integral interchange isn't fully written out, I would want an independent derivation of that exponent before relying on the Theta=1 case. The author is upfront about Assumption 1 being unsatisfactory, and the main theorem doesn't depend on it. The intermediate case 1/2 < Theta < 1 is fine because the second moment is lower order; the boundary Theta=1 is genuinely delicate. The computational classification also deserves a check, but that's a minor concern. The borderline B values in Tables 2 and 3 are close to 2 (around 1.98), so a small numerical error could flip membership in D. The method is sound, and the code availability helps, but a referee should verify a few of the borderline cases. Bottom line: Theorem 1 is solid and significant within the comparative prime number theory program. Theorem 2's Theta=1 case is a conditional side result that needs a fuller proof of Lemma 10. I'd send this to peer review and ask the referee to focus on Lemma 10 and the numerical verification.

Referee Report

2 major / 5 minor

Summary. This paper studies the averaged error terms E_i(x) in the three Mertens theorems. Theorem 1 shows, for i=1,2, that RH is equivalent to the condition that ∫_2^X E_i(x) dx > 0 for every X>2: under RH this is proved from the explicit formula for ψ and partial summation identities, while the converse uses Landau's oscillation theorem and the known numerical positivity of E_i up to 10^8. Theorem 2 treats E_3: positivity on RH when the supremun Θ of the real parts of the zeros equals 1/2, infinitely many sign changes when 1/2<Θ<1, and a corresponding conditional statement when Θ=1 under Assumption 1 and a Pintz-type oscillation lemma. Theorems 3 and 4 give analogous equivalences for real primitive Dirichlet characters and for arithmetic progressions, with finite classifications of the exceptional conductors, obtained by explicit zero computations and a verification with LCALC/SageMath.

Significance. Conditional on the gaps noted below, the paper contains a striking result: a simple positivity condition on a single integral is equivalent to RH, with no fitted parameters; the constants B1, B_chi_d, and B_q are computed from the functional equation and from zero data. The proof of Theorem 1 is essentially self-contained and uses standard tools, and the paper provides reproducible code for the computational parts. The E_3 result is more delicate and is genuinely conditional: the Θ=1 case depends on Assumption 1 and on a lemma whose proof is only sketched. If the missing details can be supplied, the paper is a solid contribution; in its present form the Θ=1 part of Theorem 2 is not proved to the same standard as the rest of the paper.

major comments (2)

[§5, Lemma 10] The proof of Lemma 10 is only a sketch. The estimate (28) is load-bearing: the denominator γ0^{2+ε} is exactly what makes −Δ1(x2) dominate Δ2(x2) in the final comparison; if the correct denominator were γ0^{1+ε}, the displayed inequalities would not imply −Δ1>Δ2. The text states that the γ1^2 factor appears from differentiation inside I2, but the interchange in (29), the power-sum argument, and the effective constants C1(ε), C2(ε) are not supplied. Since the paper explicitly adapts Pintz [17] to a different function (a double integral rather than π−li), a citation to [17] is not enough. Please give a complete proof of Lemma 10, or state it as an additional hypothesis and mark the Θ=1 clause of Theorem 2 as conditional on Lemma 10 as well.
[§5, Lemma 11] The proof of Lemma 11 is a one-sentence reduction to [18, Theorem 1]. The lemma supplies the upper bound for Δ2 that competes with Lemma 10, and the statement includes a 'finitely many zeros' caveat when Θ=1. The adaptation should be written out, at least to the extent of showing explicitly that the exceptional finite zeros do not alter the bound x/e^{2(1−ε)ω(x)}. As written, the Θ=1 case of Theorem 2 depends on two results whose proofs are not supplied at the same standard as the rest of the paper.

minor comments (5)

[§2, Eq. (19)] The positive constant 2/log2 − li(2) is dropped in the subsequent bound for π(x)−li(x); for X≥10^8 the term is negligible, but the displayed inequality should either include it or bound it explicitly.
[§4, inequality (27)] The domination of the n≥3 terms in (27) relies on |E2(x)|<1 for all x>2, citing [22, Theorem 5]. Please restate the exact range and the numerical bound from that theorem so the reader can verify the claimed inequality.
[§7, Theorem 4] The proof only handles E1; the E2 case is dismissed with 'again we only address E1 here.' Since Theorem 4 asserts the equivalence for both i=1 and i=2, the analogous identity for E2 and the analogue of (30) should be stated explicitly, together with the reason the constant B_q is the same.
[§4, proof of Lemma 9] The notation for the sums over zeros, for example 'βj ≥ 1/2 + δ', is cramped; please define the indexing set explicitly (zeros of ζ with real part β and imaginary part γ, with |γ| ≤ X^2).
[Tables 2 and 3] The tables list B_chi_d to three decimal places; the repository should include the full computed values and the exact version of LCALC used, so that the computational classification is fully reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation chain: constants are computed from explicit formulas and zero data; the Θ=1 case rests on an external theorem sketch and an explicit assumption, not on self-citation or fitted inputs.

full rationale

The main result Theorem 1 is derived from the explicit formula, Landau's oscillation theorem, and classical bounds; the constants E1, E2, and B1 are evaluated from convergent sums or known values, not fitted to the positivity conclusion. Theorem 3's set D is computed from the zero-sum criterion Bχd < 2 via an explicit zero-counting bound and LCALC, and the (b)=>(a) direction uses the nonvanishing condition L(Θχd, χd) ≠ 0, independently supplied by Platt's verification; this is classification, not circularity. The only load-bearing weakness is Theorem 2 for Θ = 1: Lemma 10 is admittedly only a sketch adapting Pintz [17], and Assumption 1 is an unproved hypothesis about the zero-free boundary. The paper flags this itself in §8(1) and in the text 'We only provide a sketch of proof as it follows the same lines as that of [17, Theorem 2]'. These are gaps in proof completeness, not instances of conclusions reducing to inputs by definition or self-citation. No fitted parameter is renamed as a prediction, no load-bearing self-citation is used, and no uniqueness claim is imported from the author's prior work.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted. The constants B1 and B_chi_d are derived from the functional equation or from known zero computations. The main additional axioms are standard analytic number theory tools, plus Assumption 1 and LI which are clearly labeled as hypotheses for specific conditional results.

assumptions (6)

standard math Explicit formula for psi(x): psi(x) = x - sum_rho x^rho/rho - log 2pi - (1/2) log(1-x^{-2}) (Lemma 5).
Invoked throughout Sections 2 and 3 to convert prime sums into zero sums.
standard math Landau's oscillation theorem for Mellin transforms with nonnegative integrands.
Used in Section 3 to force sign changes of the integral of E_i(x) when RH fails.
standard math Verified numerical inequalities from Rosser-Schoenfeld: psi(x)-theta(x) > 0.98 sqrt(x) for x >= 121 and E_i(x) > 0 for 2 <= x <= 10^8.
Used in the sufficiency proof to dominate the tails in Section 2.
standard math Ingham zero-density estimate and Grosswald's bound that psi(x)-x = O_Theta(x^Theta) when Theta > 1/2.
Used in Lemma 9 to bound the mean square of E2 when Theta > 1/2.
ad hoc to paper Assumption 1: existence of a zero-free boundary function eta satisfying properties (i)-(iv).
Needed for the Theta=1 case of Theorem 2; the paper itself states this is an assumption.
domain assumption Linear Independence conjecture for zeros of L(s,chi) (LI), or the analogous independence for products over characters modulo q.
The negative direction statements in Theorems 3 and 4 are explicitly conditional on LI.

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Pith. "Pith review of On the mean values of the error terms in Mertens' theorems." pith.science (2026). https://pith.science/paper/SGTBCBNF

@misc{pith2026241118903,
  author       = {Pith},
  title        = {Pith review of: On the mean values of the error terms in Mertens' theorems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGTBCBNF}},
  note         = {Machine review of arXiv:2411.18903}
}

abstract

For $i\in \{1,2,3\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\in \{1,2\}$ the Riemann hypothesis is equivalent to the condition $\int_2^X E_i(x) \:\mathrm{d}x>0$ for all $X>2$, and we examine assumptions under which the equivalence also holds for $i=3$. In addition, we extend our results to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.

Figures

Figures reproduced from arXiv: 2411.18903 by the authors.

**Figure 1.** A plot of f1(X) and f2(X) for 2000 ≤ X ≤ 106 . Note that at X = 106 they are still far above their limsup. zeros of ζ(s) are linearly independent over the rationals (note that non-simple zeros are permitted here), then the first inequality in (2) can be strengthened to an equality. From (2) it is not hard to derive a localized version of Theorems 1 and 2: Corollary 1. Assuming RH, for each i ∈ {1, 2, 3} and any pos… view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [17]

On the remainder term of the prime number formula I. On a prob- lem of Littlewood

J. Pintz. “On the remainder term of the prime number formula I. On a prob- lem of Littlewood”. In: Acta Arith. 36(4) (1980), pp. 341–365

work page 1980

[2] [1]

Counting zeros of Dirichlet L-functions

M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer. “Counting zeros of Dirichlet L-functions”. In: Math. Comp. 90(329) (2021), pp. 1455–1482

work page 2021

[3] [2]

The mean square of the er- ror term in the prime number theorem

R. P. Brent, D. J. Platt, and T. S. Trudgian. “The mean square of the er- ror term in the prime number theorem”. In: J. Number Theory 238 (2022), pp. 740–762

work page 2022

[4] [3]

On the first sign change in Mertens’ theorem

J. B¨ uthe. “On the first sign change in Mertens’ theorem”. In: Acta Arith. 171 (2) (2015), pp. 183–195

work page 2015

[5] [4]

An analytic method for bounding ψ(x)

J. B¨ uthe. “An analytic method for bounding ψ(x)”. In: Math. Comp. 87 (312) (2018), pp. 1991–2009

work page 2018

[6] [5]

Davenport

H. Davenport. Multiplicative Number Theory . 2nd ed. Graduate Texts in Mathematics 74. Springer Verlag, New York, 1980

work page 1980

[7] [6]

Sagemath, the Sage Mathematics Software System (Version 10.4)

Sage Developers. Sagemath, the Sage Mathematics Software System (Version 10.4). https://www.sagemath.org. 2024

work page 2024

[8] [7]

Oscillation of Mertens’ product formula

H. G. Diamond and J. Pintz. “Oscillation of Mertens’ product formula”. In: Th´ eor. Nombres Bordeaux21(3) (2009), pp. 523–533

work page 2009