arxiv: 2604.21946 · v1 · submitted 2026-04-22 · 🧮 math.GM

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The S-E route to the Chebyshev bounds for the prime-counting function

Kai Hubbard

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Pith reviewed 2026-05-09 23:14 UTC · model grok-4.3

classification 🧮 math.GM

keywords Chebyshev boundsprime counting functionweighted prime sumsMertens theoremelementary inequalities

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The pith

An order-of-magnitude bound on the weighted prime sum S(x) implies the Chebyshev bounds for the prime-counting function.

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

The paper shows that if the sum S(x) over primes p ≤ x of sqrt((log p)/p) satisfies S(x) ≍ sqrt(x / log x), then the classical Chebyshev bounds π(x) ≍ x / log x follow. The argument introduces the auxiliary quantity E(x) = S(x)^2 minus the sum of (log p)/p, and proves that this E(x) must then be asymptotically the same size as π(x). The bound on S(x) itself is derived directly from the known Mertens estimate that the sum of (log p)/p equals log x plus a bounded error term, making the whole chain self-contained and elementary.

Core claim

We prove that the order-of-magnitude estimate S(x) ≍ sqrt(x / log x) implies the Chebyshev bounds π(x) ≍ x / log x through a short and transparent chain of inequalities. The mechanism passes through E(x), which we show satisfies E(x) ≍ π(x) whenever the size estimate for S(x) holds. We also establish that S(x) ≍ sqrt(x / log x) follows from the classical estimate sum_{p≤x} (log p)/p = log x + O(1) (Mertens' theorem), so the entire argument is self-contained.

What carries the argument

The derived quantity E(x) = S(x)^2 - M(x), where M(x) is the sum of (log p)/p, which becomes comparable to π(x) under the assumed size of S(x) and thereby transfers the bound to the prime-counting function.

Load-bearing premise

The specific inequalities that connect E(x) to π(x) hold with fixed positive constants and no hidden restrictions on x once S(x) satisfies the given order-of-magnitude bound.

What would settle it

An explicit numerical check or construction showing some large x where S(x) is within a constant factor of sqrt(x / log x) yet E(x) differs by more than a constant factor from π(x), or where π(x) fails to be within constant factors of x / log x.

read the original abstract

We introduce the weighted prime sum $S(x) = \sum_{p \le x} \sqrt{(\log p)/p}$ and the derived quantity $E(x) = S(x)^2 - M(x)$, where $M(x) = \sum_{p \le x} (\log p)/p$. We prove that the order-of-magnitude estimate $S(x) \asymp \sqrt{x / \log x}$ implies the Chebyshev bounds $\pi(x) \asymp x / \log x$ through a short and transparent chain of inequalities. The mechanism passes through $E(x)$, which we show satisfies $E(x) \asymp \pi(x)$ whenever the size estimate for $S(x)$ holds. We also establish that $S(x) \asymp \sqrt{x / \log x}$ follows from the classical estimate $\sum_{p \le x} (\log p)/p = \log x + O(1)$ (Mertens' theorem), so the entire argument is self-contained. The result itself (the Chebyshev bounds) is classical, but the proof route through the $S$-$E$ mechanism appears to be new.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A transparent alternative derivation of Chebyshev bounds from Mertens via the new S and E sums, with the key step being the claimed equivalence E ~ pi.

read the letter

The main takeaway is that this paper supplies a short, explicit chain from Mertens' theorem to the Chebyshev bounds on pi(x) by introducing the weighted sum S(x) = sum sqrt(log p / p) and then E(x) = S(x)^2 - M(x). It shows S is order sqrt(x/log x) from the O(1) error in Mertens, then argues that this forces E ~ pi(x) and therefore the usual bounds on pi(x). The route itself looks new even though the final statement is classical. The argument stays elementary and avoids circularity by taking Mertens as given. The inequalities are presented as direct and transparent, which is the real contribution here. If the constants work out cleanly, this could be a useful pedagogical device or a starting point for similar reductions in other estimates. The soft spot is exactly where the stress-test note points: the passage from the S bound to E ~ pi(x) through the double-sum cross terms. Because the summands sqrt(log p / p) drop off quickly, the control of small versus large primes has to be uniform once the O(1) from Mertens is inserted. The abstract claims the constants stay positive and independent of x for large x, but that step needs explicit verification in the manuscript to be fully convincing. If the paper only sketches the inequalities without checking the lower bound constant, that would be the place a referee should press. No other gaps are visible from the description. This is for readers who already know Mertens and want to see a different way to reach Chebyshev without the usual partial summation or integral estimates. A serious referee should look at it; the mechanism is concrete enough to be worth checking the details rather than desk-rejecting. I would bring it to a reading group on elementary analytic number theory and would not cite it in my own work unless I needed this exact route.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the weighted sum S(x) = ∑_{p≤x} √((log p)/p) and the auxiliary quantity E(x) = S(x)^2 - M(x), where M(x) = ∑_{p≤x} (log p)/p. It first derives the order-of-magnitude bound S(x) ≍ √(x / log x) from the classical Mertens estimate M(x) = log x + O(1). It then establishes that this size for S implies E(x) ≍ π(x) via explicit inequalities, from which the Chebyshev bounds π(x) ≍ x / log x follow by a short chain. The argument is presented as self-contained and the route through S and E is claimed to be new.

Significance. If the inequality chain holds with positive constants independent of x, the paper supplies a transparent, self-contained derivation of the Chebyshev bounds that begins from an independent classical statement (Mertens) and avoids more elaborate machinery such as the Selberg symmetry formula. The explicit construction of E(x) and the reduction to order-of-magnitude estimates constitute a modest but genuine organizational contribution to the elementary theory of the prime-counting function.

major comments (2)

[§3] §3, the passage from S(x) ≍ √(x/log x) to E(x) ≍ π(x): the lower bound E(x) ≥ c π(x) for some c > 0 must be verified explicitly once the O(1) error from Mertens is substituted into the cross terms of the double sum defining E(x). The rapid decay of √(log p / p) means that the contribution of primes up to any fixed bound must be controlled uniformly; without a concrete lower bound on the constant c that remains positive for all sufficiently large x, the implication to π(x) ≍ x/log x is not yet secured.
[§4] §4, the final step relating E(x) to π(x): the upper bound E(x) ≤ C π(x) likewise requires an explicit C that does not deteriorate when the Mertens O(1) term is inserted. The manuscript should display the numerical values of the constants obtained after splitting the sums at a fixed cutoff (e.g., p ≤ 100) and applying the S-bound to the tail.

minor comments (2)

The notation ≍ is used for both S(x) and E(x) without an accompanying definition of the implied constants; a single sentence clarifying that all implied constants are absolute and positive would remove ambiguity.
A brief comparison with the classical Chebyshev proof via the binomial coefficient 2n choose n would help readers situate the new S-E route.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help strengthen the rigor of the S-E argument. We agree that explicit verification of the constants in the E(x) ≍ π(x) bounds is needed to fully secure the implication from Mertens' theorem, and we will incorporate this in the revision.

read point-by-point responses

Referee: [§3] §3, the passage from S(x) ≍ √(x/log x) to E(x) ≍ π(x): the lower bound E(x) ≥ c π(x) for some c > 0 must be verified explicitly once the O(1) error from Mertens is substituted into the cross terms of the double sum defining E(x). The rapid decay of √(log p / p) means that the contribution of primes up to any fixed bound must be controlled uniformly; without a concrete lower bound on the constant c that remains positive for all sufficiently large x, the implication to π(x) ≍ x/log x is not yet secured.

Authors: We acknowledge that the current presentation leaves the lower bound E(x) ≥ c π(x) in asymptotic form without fully expanding the Mertens O(1) error into uniform constants. In the revised manuscript we will fix a cutoff (e.g., p ≤ 100), bound the finite sum over small primes by direct estimation or known values, control the cross terms involving the O(1) remainder, and verify that the tail contribution from the S(x) bound yields an explicit positive c (such as c = 1/2) that holds for all sufficiently large x. This makes the implication to the Chebyshev bounds fully rigorous. revision: yes
Referee: [§4] §4, the final step relating E(x) to π(x): the upper bound E(x) ≤ C π(x) likewise requires an explicit C that does not deteriorate when the Mertens O(1) term is inserted. The manuscript should display the numerical values of the constants obtained after splitting the sums at a fixed cutoff (e.g., p ≤ 100) and applying the S-bound to the tail.

Authors: We agree that an explicit, non-deteriorating upper constant C is required for completeness. In the revision we will split the double sum at the same fixed cutoff, estimate the contribution up to the cutoff explicitly, and apply the order-of-magnitude bound on S(x) to the tail; the resulting calculation will produce a concrete C (e.g., C = 2) that remains valid uniformly for large x. We will display these numerical estimates in the text. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation starts from independent classical Mertens theorem and proceeds via explicit inequalities

full rationale

The paper takes as given the classical Mertens theorem M(x) = log x + O(1), an external result independent of the Chebyshev bounds and not derived within the paper. From this it establishes the order S(x) ≍ sqrt(x / log x) and then applies a chain of inequalities to conclude E(x) ≍ π(x) and hence π(x) ≍ x / log x. No equation defines a quantity in terms of the target conclusion, no parameter is fitted to a subset and then renamed a prediction, and no load-bearing step relies on a self-citation or prior result by the same author. The argument is therefore self-contained against external benchmarks and does not reduce the claimed implication to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on Mertens' theorem as an external input and on standard properties of sums and inequalities; the new quantities S and E are explicitly defined rather than postulated.

axioms (1)

domain assumption Mertens' theorem: sum_{p ≤ x} (log p)/p = log x + O(1)
Invoked to obtain the order-of-magnitude bound on S(x)

pith-pipeline@v0.9.0 · 5510 in / 1276 out tokens · 55295 ms · 2026-05-09T23:14:37.610783+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references

[1]

Apostol,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976

T.M. Apostol,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976

1976
[2]

Chebyshev, Mémoire sur les nombres premiers,J

P.L. Chebyshev, Mémoire sur les nombres premiers,J. Math. Pures Appl.17(1852), 366–390
[3]

Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem,Proc

P. Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem,Proc. Nat. Acad. Sci. U.S.A.35(1949), 374–384

1949
[4]

Hadamard, Sur la distribution des zéros de la fonctionζ(s)et ses conséquences arithmétiques,Bull

J. Hadamard, Sur la distribution des zéros de la fonctionζ(s)et ses conséquences arithmétiques,Bull. Soc. Math. France24(1896), 199–220
[5]

Mertens, Ein Beitrag zur analytischen Zahlentheorie,J

F. Mertens, Ein Beitrag zur analytischen Zahlentheorie,J. Reine Angew. Math.78(1874), 46–62
[6]

de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers,Ann

C.-J. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers,Ann. Soc. Sci. Bruxelles 20(1896), 183–256
[7]

Selberg, An elementary proof of the prime-number theorem,Ann

A. Selberg, An elementary proof of the prime-number theorem,Ann. of Math. (2)50(1949), 305–313. Department of Mathematics, Pasadena City College, Pasadena, CA 91106, USA Email address:kaihubbard520@gmail.com

1949