On the connection between zero-free regions and the error term in the Prime Number Theorem

Frederik Broucke

On the connection between zero-free regions and the error term in the Prime Number Theorem

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Pith's one-line read A zero-free region for a Beurling zeta function forces the prime-counting error term into one explicit formula, and matching examples show the formula is sharp up to a constant.

desk verdict Sharp refinement of the Pintz–Révész connection for regularly and slowly varying zero-free regions, with a matching Beurling construction; solid and worth refereeing. read the letter →

arxiv 2507.13780 v1 pith:L4OQGIR2 submitted 2025-07-18 math.NT

Frederik Broucke This is my paper · ORCID

classification math.NT MSC 11M2611N0511M4111N80

keywords zero-freeregionsprimenumbertheoremwithremainderBeurlinggeneralizedsystemszetafunctionsoscillationoftheerrortermzero-densityestimatesregularlyvaryingsharpnessconstructions

topics The Riemann Hypothesis 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

This paper answers, in the setting of generalized (Beurling) primes, the old question of what remainder term in the prime number theorem follows from a prescribed zero-free region. The first half proves that a zero-free region $\sigma>1-f(\log|t|)$, with $f$ eventually $C^1$, decreasing, strictly convex, and regularly or slowly varying, forces $\psi_P(x)-x\ll_\delta x\exp(-\omega(x)+(A+\delta)f(u_0(x))^B u_0(x))\,\omega(x)^C$ whenever the zeta function satisfies a zero-density estimate of the usual shape; here $\omega(x)$ is the minimum of $f(u)\log x+u$ and $u_0(x)$ is the unique minimizer. The second half constructs, for every such $f$, a Beurling number system whose zeta function has infinitely many zeros exactly on the contour $\sigma=1-f(\log|t|)$ and none to its right, and whose error term oscillates at the matching scale $x\exp(-\omega(x)+(1-\delta)f(u_0(x))u_0(x))$. Together the two directions identify the exponential scale $x e^{-\omega(x)}$, with a lower-order correction $f(u_0)u_0$, as the true, essentially sharp connection between zero-free regions and prime-counting error.

What carries the argument

The carrying object is the minimizer pair $(u_0(x),\omega(x))$: with $h(u,x)=f(u)\log x+u$, strict convexity gives a unique minimizer $u_0(x)$, and $\omega(x)=h(u_0(x),x)$ is the exponential decay scale of the error term. The upper-bound proof splits zeros at heights $\gamma=\exp(u_0(x))$, using the zero-density estimate to control how many zeros lie where the exponential weight $e^{-h(\log\gamma,x)}$ is non-negligible. For the reverse direction, the paper plants zeros by multiplying the zeta function by factors $G(a(s-\rho))$ with $G(z)=(1-e^{-z}-e^{-2z})/z$, an entire function whose logarithm is the Mellin transform of a nonnegative, compactly supported function, so an infinite sequence of zeros can be introduced while keeping the generalized-prime counting function nondecreasing. The zeros are arranged in blocks near heights $\gamma_k=\exp(4^k)$ with real part $1-1/\ell_k$, $\ell_k=1/f(\log\gamma_k)$, so that their explicit-formula contributions interfere constructively at a selected sequence $x_k$ and destructively in the variant leading to Proposition 5.2.

What would settle it

For $f(u)=u^{-1/2}$, the construction predicts a system whose dominant explicit-formula terms at heights $\gamma_k=\exp(4^k)$ add up near $x_k=u_0^{-1}(4^k)$ to oscillations of order $x_k\exp(-\omega(x_k)+(\log x_k)^{1/3})$; computing these finite sums for large $k$ and checking that their signs alternate at the predicted amplitude would confirm the mechanism, while bounded or much smaller oscillations would falsify the interference claim of Theorem 1.6.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the map from zero-free regions to prime-number-theorem error terms is captured, up to the value of one constant, by the variational quantity $\omega(x)$. Theorem 3.1 shows that for a Beurling zeta function satisfying Axiom A (generalized integers counted by $Ax+O(x^\theta)$) and a zero-density estimate $N(\sigma,T)\ll T^{A(1-\sigma)^B}(\log T)^C$, any zero-free region $\sigma>1-f(\log|t|)$ inside the regular/slow variation class forces the explicit upper bound displayed in the summary; with the currently best Beurling zero-density estimate this becomes the concrete bound of Theorem 1.5. Theorem 1.6 shows the converse construction: for every admissible $f$ with $1/u=o(f(u))$, there is a Beurling number system with $N_P(x)=Ax+O_\varepsilon(x^{1/2+\varepsilon})$, with infinitely many zeros on $\sigma=1-f(\log|t|)$ and none to the right, and with $\psi_P(x)-x=\Omega_\pm(x\exp(-\omega(x)+(1-\delta)f(u_0(x))u_0(x)))$ for every $\delta>0$. The paper also proves a destructive-interference variant showing the parameter $\varepsilon$ in the oscillation theorem cannot be set to zero, and identifies that the classical method for deriving zero-free regions from growth is sharp up to a constant among extended Beurling systems.

Load-bearing premise

Everything in the fully sharp results rests on the contour function $f$ being eventually smooth, decreasing, strictly convex, and either regularly varying of index $-\alpha$ or slowly varying with the extra technical conditions (1.9)--(1.10); outside this class the paper does not claim the refined bounds.

Editorial extensions

If this is right

For the classical zeta function, the theorem applies to the asymptotically best known zero-free region, whose $f$ is regularly varying of index $-2/3$, together with existing zero-density estimates; in that situation the exponent $C$ in the density estimate matters more than the constant $A$.
For Beurling systems, the constructed examples prove that a prescribed zero-free region alone does not force any improvement over the Carlson-type zero-density estimate: one can have $N(\sigma,T)=\Omega(T^{(2b-1-o(1))f(\log T)})$ while still being zero-free to the right of $1-f(\log|t|)$.
The upper and lower bounds together imply that the epsilon in the earlier epsilonic theorems can be replaced by a function tending to zero, but not by zero: Proposition 5.2 constructs systems with zeros exactly on $\sigma=1-(\log|t|)^{-\alpha}$, none to the right, and error term $O(xe^{-\omega(x)}(\log x)^{-1/(2(1+\alpha))})$.
The analysis in Section 6 shows that the classical method of deducing a zero-free region from an upper bound on the zeta function is sharp, up to a universal constant, within the class of extended Beurling systems.

Reading between the lines

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

Beyond the paper: if the same sharpness transfers to zeta functions with a fixed zero distribution, then any future narrowing of the zero-free region would automatically force a prime-counting error term of shape $x\exp(-\omega(x)+O(f(u_0)u_0))$; the remaining target would be removing the $\omega(x)^C$ factor.
The pair of constructive and destructive interference constructions suggests that the error term is controlled not merely by the boundary of the zero-free region but by correlations among nearby zeros; one could test this by computing, for finite truncations of the constructed systems, how much of the oscillation survives when the zero block near $x_k$ is randomly perturbed.
A natural variational problem suggested by Theorem 1.6 is to minimize, over all Beurling systems with a fixed zero-free contour, the constant in front of $f(u_0)u_0$ in the oscillation; the construction shows the infimum is at most $1$, while the upper-bound proof only gives the constant $A$.

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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

Read the full paper. The main new things are Theorem 1.5 and Theorem 1.6. The first improves Johnston's recent result for the natural class of regularly or slowly varying f and extends it from the Riemann zeta function to general Beurling systems, replacing the epsilon loss with a secondary term of size f(u0)^B u0 times omega^C. The second generalizes the Diamond–Montgomery–Vorhauer construction from f(u) = c/u to arbitrary such f, plants zeros exactly on the prescribed contour with none to the right, and gets an oscillation lower bound that matches the upper bound up to the constant. That is a genuine step forward on Ingham's question.

The upper-bound proof is a careful adaptation of the Pintz–Révész template with dyadic splits; I found the estimates coherent. The construction in Section 5 verifies positivity of psi_c', bounds the zeta function, locates the zeros, and derives the oscillation estimates; the random approximation to pass to a discrete system is standard and correctly applied. I agree with the reader's ACCEPT verdict and with the stress-test note: no load-bearing flaw.

Soft spots, in proportion. The theorems are restricted to f that is eventually C^1, decreasing, strictly convex, and regularly or slowly varying. The author flags this as the most restrictive assumption, and it is what makes u0(x) unique and gives (2.3)–(2.4). Outside that class you only get Révész's epsilonic versions, so the scope is delimited but not internally broken. Two small presentational issues in the written proof: in Theorem 3.1 the S1 and S3 estimates invoke the concrete zero-density estimate (3.3) rather than the general hypothesis (3.2), though the same epsilon-absorption works under (3.2); and in Section 5.5 the displayed asymptotic f(u0(x_t)) ~ (1+alpha) omega/log x is a typo for the reverse ratio. Neither affects the claims.

The self-citations of [5] and [7] are fine: those are independent published results and do not smuggle in the target error-term bound.

Who this is for: analytic number theorists working on Ingham's problem, Beurling generalized primes, and zero-free regions. It deserves a serious referee; I would send it to a specialist and expect the main theorems to stand. Minor revisions should fix the general-hypothesis citation and the typo.

Referee Report

0 major / 3 minor

Summary. The paper studies the relation between zero-free regions of Beurling zeta functions and the error term in the prime number theorem. Assuming Axiom A (1.1) and a zero-density estimate (3.2), Theorem 3.1 gives an upper bound for ψP(x)−x of the form x exp(−ω(x) + (A+δ)f(u0(x))^B u0(x)) ω(x)^C for zero-free regions σ > 1 − f(log t), where f is eventually C^1, decreasing, strictly convex, and either regularly varying of index −α (0<α≤1) or slowly varying with conditions (1.9)–(1.10). This refines the ε-loss in Pintz–Révész to a decaying secondary term. Theorem 1.6 provides a converse construction: for every such f with 1/u = o(f(u)), there exists a Beurling number system whose zeta function has infinitely many zeros on the contour σ = 1 − f(log|t|), none to the right, and whose PNT error oscillates at the complementary level x exp(−ω(x) + (1−δ)f(u0(x))u0(x)). The paper also includes variants: a Lindelöf-type construction with only xe^{−ω(x)} oscillation, an example showing ε cannot be zero in the converse Pintz theorem, and remarks on Landau's method and zero clustering.

Significance. The main results give the sharpest known quantitative connection between zero-free regions and PNT error terms in the Beurling setting, improving on Pintz, Johnston, and Révész by removing the arbitrary ε in the exponent at the cost of a secondary term. The constructive half of the paper is a substantial technical achievement: it verifies positivity of the prime measure, bounds the resulting zeta function, locates all zeros, and derives matching oscillation estimates, building on the Diamond–Montgomery–Vorhauer method and the author's random approximation theorem. The paper also gives useful structural insights, showing that zero-free regions alone do not imply improved zero-density estimates, and that Landau's method is nearly sharp for a class of Beurling zeta functions. The restriction to regularly or slowly varying f is explicitly acknowledged and covers essentially all zero-free regions used in practice.

minor comments (3)

[Section 3, proof of Theorem 3.1] The estimates for S1 and S3 use the concrete zero-density estimate (3.3) with exponents 5ε/(1−θ), while the theorem is stated under the general hypothesis (3.2). The proof should be modified to use (3.2) directly, choosing ε sufficiently small relative to δ and absorbing the logarithmic factor; this is a presentational issue, but as written the proof does not cover the generality claimed.
[Section 3, first paragraph after the definition of S2] The sentence "Suppose first that f is slowly varying of index −α, 0 < α ≤ 1" should read "regularly varying" because the subsequent use of (2.3) is the regular-variation asymptotics.
[Section 5.5, displayed relation before the final comparison] The approximation f(u0(xt)) ∼ (1+α) ω(xt)/log xt is reversed; (2.3) gives f(u0(xt)) ∼ ω(xt)/((1+α) log xt). The intended conclusion f(u0(xt)) ∼ f(u0(xk)) is unaffected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the upper-bound theorem uses independent zero-density inputs and the lower-bound construction is an explicit example, not a disguised fit.

full rationale

The paper's derivation chain is self-contained and non-circular. Theorem 1.5/3.1 takes as inputs an assumed zero-free region sigma > 1 - f(log|t|), an explicit Riemann-von Mangoldt formula (3.1) from Revesz, and a general zero-density estimate (3.2); the output error term is obtained by bounding the zero sum via h(log gamma, x) >= omega(x) and by using the zero-density estimate only to control the number of zeros in thin strips. The zero-density estimate (3.3) is cited from the author's prior work [5], but it is an independent published theorem that does not contain the target error-term bound, and Theorem 3.1 is stated for an arbitrary estimate (3.2), so the self-citation is not load-bearing in a circular sense. The random approximation theorem [7] used in Theorem 1.6 is likewise an independent external tool. The lower-bound construction explicitly plants zeros on the contour sigma = 1 - f(log|t|) via the Diamond-Montgomery-Vorhauer factors G(a_k(s-rho_{k,m})) and then computes the oscillation of psi_c from the explicit formula; the choice of x_k = u_0^{-1}(log gamma_k) and c_{k,m} to align phases is a construction of an example, not a prediction fitted to data. There is no equation in which the conclusion is definitionally equal to an input, no fitted parameter renamed as a prediction, and no uniqueness theorem imported from the author's prior work to force the choice. The acknowledged restriction to regularly or slowly varying f (Section 2) is a scope limitation, not a circular step.

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

No empirical data are fitted; the free-parameter list is empty because the proof parameters (epsilon, delta, k0, scale ratio 4) are generic choices, not tuned constants. The axioms are standard analytic number theory ingredients and prior published theorems. No new physically motivated entities are introduced; the constructed Beurling systems are explicit mathematical objects.

assumptions (6)

domain assumption Axiom A (1.1): N_P(x) = Ax + O(x^theta) for some A > 0 and theta in [0,1).
Defines the class of Beurling systems studied; used in the explicit formula (3.1), the zero-density estimate (3.3), and throughout Sections 3 and 5.
domain assumption f is eventually C^1, decreasing, strictly convex, and either regularly varying of index -alpha (0 < alpha <= 1) or slowly varying satisfying (1.9) and (1.10).
Assumed in Theorems 1.5 and 1.6; guarantees uniqueness of u0(x) and the asymptotic relations (2.3) and (2.4) used to bound h(u,x). Section 2 states this is the most restrictive assumption.
domain assumption Zero-density estimate (3.3): N(sigma,T) << T^{4(1-sigma)/(1-theta)} (log T)^9, uniformly for sigma >= (1+theta)/2.
Quoted from the author's paper [5]; fixes A = 4/(1-theta) and C = 9 in Theorem 1.5. Not proved in this preprint.
domain assumption Explicit Riemann-von Mangoldt formula (3.1) for psi_P, quoted from Revesz [21,23] with a correction.
Converts prime-counting error into a sum over zeta zeros; is the backbone of both the upper bound and the oscillation construction.
domain assumption The Diamond-Montgomery-Vorhauer function G(z) = 1 - e^{-z} - e^{-2z}/z has the Mellin representation and properties summarized in Lemma 4.1.
Used to plant zeros at prescribed points while keeping the prime measure nondecreasing; properties are taken from [8] and [9, Section 17.6].
domain assumption Random approximation theorem of Broucke-Vindas (Theorem 5.1) converts an extended system into a discrete system with O(1) prime-count error.
Crucial for transferring zeros and oscillation from zeta_c to an actual Beurling prime system; quoted from the author's prior paper [7].

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Cite this review

Pith. "Pith review of On the connection between zero-free regions and the error term in the Prime Number Theorem." pith.science (2026). https://pith.science/paper/L4OQGIR2

@misc{pith2026250713780,
  author       = {Pith},
  title        = {Pith review of: On the connection between zero-free regions and the error term in the Prime Number Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4OQGIR2}},
  note         = {Machine review of arXiv:2507.13780}
}

read the original abstract

We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and R\'ev\'esz (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

Broucke, On zero-density estimates for Beurling zeta functions , to appear in Ann

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work page 2024

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work page Pith review arXiv 2023

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work page 2024

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Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers g´ en´ eralis´ es

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