On the arithmetic average of the first $n$ primes

Matt Visser (Victoria University of Wellington)

REVIEW 3 major objections 4 minor 1 cited by

On the arithmetic average of the first $n$ primes

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The arithmetic average of the first $n$ primes has gaps between $(\ln n)/2$ and $\ln n$, so suitably formulated prime-averaged analogues of nine classic conjectures are provable theorems, not conjectures.

desk verdict True and repairable, but the printed proof of the central lower bound has a real algebra error. read the letter →

arxiv 2505.04951 v3 pith:COY2HEYN submitted 2025-05-08 math.NT

Matt Visser (Victoria University of Wellington) This is my paper · ORCID

classification math.NT MSC 11N0511A41

keywords nthprimeaverageoffirstnprimesgapsprime-averagedconjecturesCramerconjectureAndricasmoothingexplicitestimates

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

This paper studies $\bar p_n$, the average of the first $n$ primes, and proves that its successive gaps are tiny: $(\ln n)/2 < \bar g_n < \ln n$ for $n\ge 3$. Because the gaps are so small, the averaged sequence satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures, and these analogues are theorems rather than conjectures. The paper also notes that although $\bar p_n$ and $p_n$ carry identical information through the inversion $p_n = n\bar p_n - (n-1)\bar p_{n-1}$, the averaging process smooths away the wild local fluctuations of ordinary primes. The original conjectures for ordinary primes remain untouched: the link between ordinary and averaged gaps is too weak to transfer the new bounds back.

What carries the argument

The central object is the averaging map $\bar p_n = \frac{1}{n}\sum_{i=1}^n p_i$, together with the derived gap bound $(\ln n)/2 < \bar g_n < \ln n$. The map turns the $n$th prime into an arithmetic mean, and its key property is smoothing: although $\bar p_n \sim p_n/2$ asymptotically, the averaged gaps shrink like $\ln n$ in absolute terms and like $2/n$ relative to $\bar p_n$, whereas ordinary prime gaps are highly irregular. All the averaged-prime theorems follow from this gap bound combined with explicit inequalities for $p_n$, $\bar p_n$, and $\pi(x)$; the only recurring difficulty is designing convincing analogues of each original conjecture.

What would settle it

Compute $\bar p_n = (\text{sum of first }n\text{ primes})/n$ exactly and compare $\bar g_n = \bar p_{n+1}-\bar p_n$ with $\ln n$. A single verified case with $n\ge 3$ where $\bar g_n\ge\ln n$, or $n\ge 1$ where $\bar g_n\le(\ln n)/2$, would refute the central bound; scanning up to a large computable range, say $n=10^6$, is a direct test, and a violation of the stated ratio window $0.7/n<\bar g_n/\bar p_n<2/n$ for $n\ge 4$ would also falsify the claim.

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Extended reading notes

Core claim

The paper's central claim is that the averaged-prime gap $\bar g_n = \bar p_{n+1} - \bar p_n$ satisfies $(\ln n)/2 < \bar g_n < \ln n$ for $n\ge 3$, and consequently $\bar g_n/\bar p_n$ is bounded between roughly $0.7/n$ and $2/n$ for $n\ge 4$. This extremely fast relative decay is the mechanism that lets the author prove averaged-prime analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. The proof combines standard explicit estimates for primes and prime-counting functions with finite checks below explicit thresholds: upper bounds on $p_n$, lower bounds on $\bar p_n$, the inequality $\bar p_n < p_n/2$, and Dusart-type estimates for $\pi(x)$. The author emphasizes that the result does not transfer back to ordinary primes, since the exact identity relating $g_n$ to $\bar g_n$ yields only $g_n = O(n\ln n)$.

Load-bearing premise

The entire proof depends on published explicit bounds for primes and prime-counting functions being correct in the ranges used, and on the listed finite checks for small $n$ being error-free; if any of those fail, the gap bounds and every averaged analogue could fail.

Editorial extensions

If this is right

For $n\ge 3$, $\bar g_n < \ln n$ implies the averaged-prime analogue of Cramer's conjecture, $\bar g_n = O((\ln \bar p_n)^2)$, holds trivially.
The averaged-prime analogue of Andrica's conjecture holds, and in fact $\sqrt{\bar p_{n+1}} - \sqrt{\bar p_n} < \sqrt{\ln n/n}$ for $n\ge 2$.
The averaged-prime analogue of Legendre's conjecture holds for real $m\ge 1$: there is at least one averaged prime between consecutive squares, and for $m\ge 1$ the count difference is at least $(2m-1)/(2\ln(m+1))$.
The averaged-prime analogue of Firoozbakht's conjecture holds: $(\bar p_n)^{1/n}$ is decreasing for all $n\ge 1$, and the Fourges, Nicholson, and Farhadian analogues follow on their stated ranges.
These results apply only to the averaged sequence; the paper's inversion identity is too weak to prove any of the original conjectures for ordinary primes.

Reading between the lines

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

This suggests a testable general principle: a monotone sequence whose gaps are $O(\ln n)$ with relative gaps $O(1/n)$ will satisfy analogues of these conjectures; one could probe the idea by averaging other sparse integer sequences, such as squarefree numbers or primes in arithmetic progressions.
The proof's finite checks (up to roughly $n=440$ and the first 55 averaged primes) could be independently re-verified with exact arithmetic, and the method would be strengthened if the bounds could be extended inductively from a computable threshold without case checking.
The failure of the back-transfer suggests the original prime conjectures are governed by local fluctuations that averaging removes; the averaged analogues are therefore best read as illustrations of smoothing rather than as evidence for the ordinary conjectures.

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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

Colleague,

You should know two things about this paper. First, the main theorem — that averaged-prime analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Nicholson, and Farhadian are all provable — is true, and the key bound \bar g_n = Θ(ln n) is the right way to see it. Second, the proof of the lower half of that bound as printed is wrong: Eq (30) contains an algebra slip, so the displayed derivation of \bar g_n > (ln n)/2 for all n ≥ 1 does not go through. The stress-test note is correct on this. A repair exists using Hassani's stronger Mandl inequality (\bar p_n < p_n/2 − n/14 for n ≥ 10), which gives the bound for n ≥ 10, with small n checked separately. So the result is salvageable and almost certainly true, but the paper as written has a gap in its central estimate.

What is genuinely new: the averaged sequence \bar p_n is smooth enough that all these famous conjectures become theorems, and the gap bound \bar g_n = Θ(ln n) is the engine. Section 6 is honest that this does not transfer back to the original prime gaps; naive inversion only gives g_n = O(n ln n). That is the right thing to say, and it keeps the paper from overclaiming.

The soft spots: besides Eq (30), Eq (87) claims an inequality holds for n ≥ 5 that actually fails for n = 5..10 (though the final domain n ≥ 7 is verified by a finite check). The paper repeatedly says 'explicitly checking smaller integers' without showing the table, which is annoying for reproducibility but minor. The proofs are short consequences of Rosser–Schoenfeld, Dusart, Mandl, and Hassani; no free parameters, no circularity. The analogues are somewhat ad hoc — the author admits the main difficulty is designing them — but that is a feature, not a flaw.

This is a paper for people who like clean smoothing arguments and want to see how averaging tames prime gaps. It is not a breakthrough about the primes themselves, and the author does not pretend it is. With the two errors fixed and the finite checks documented, it would be a solid contribution.

My call: send it to peer review if it comes your way, but require the corrections. I would not desk-reject it; the central observation is valid and the exposition is mostly clear. I won't be citing it in my own work unless I work on averaged prime sequences, but it deserves a serious referee.

Best,

Referee Report

3 major / 4 minor

Summary. The paper defines the arithmetic average of the first n primes, \bar p_n = (1/n)\sum_{i=1}^n p_i, and the associated counting function \bar\pi(x) = \#\{i : \bar p_i \le x\}. The main technical result is a pair of bounds on the averaged-prime gaps \bar g_n = \bar p_{n+1}-\bar p_n: the paper claims \bar g_n < \ln n for n\ge 3 and \bar g_n > (\ln n)/2 for n\ge 1, which imply \bar g_n/\bar p_n is bounded by constants over n. Using these bounds and explicit estimates of Rosser--Schoenfeld, Mandl, Dusart, and Hassani, the paper derives averaged-prime analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures, showing that these analogues are theorems rather than conjectures. The closing discussion notes that the averaged results do not transfer directly to the ordinary primes.

Significance. If the proof gaps identified below are repaired, the paper makes a clean and useful observation: averaging the primes smooths out local fluctuations so strongly that several classical conjectural gap and spacing statements become provable theorems for the averaged sequence. A particular strength is the explicit use of published estimates, which makes the argument largely quantitative and leaves only finite checks to be documented. The central idea—that the invertible transformation p_n \leftrightarrow \bar p_n can convert conjectures into theorems after averaging—is novel and of interest to number theorists, even though the paper correctly cautions that the averaged results do not resolve the original conjectures.

major comments (3)

[Sec. 4, Eq. (30)] The displayed chain contains an algebra error: from p_{n+1} \ge p_n+2 and \bar p_n < p_n/2 one obtains \bar g_n > (p_n/2+2)/(n+1), not (p_n/2+4)/(n+1). Using p_n>n\ln n then gives \bar g_n > (n\ln n/2+2)/(n+1), whose difference from \ln n/2 is (2-\ln n/2)/(n+1), which is negative for n>e^4. Thus Eq. (31) and the lower bound in Eq. (32) are not proved as written. A repair exists: combining Eq. (6) with p_{n+1} \ge p_n+2 yields \bar g_n > (p_n/2+2+n/14)/(n+1) > (n\ln n/2+2+n/14)/(n+1), which exceeds \ln n/2 for n\ge 10, with n\le 9 checked separately. This correction must be made and the consequences for later sections, especially the Brocard analogue, re-verified.
[Sec. 5.5, Eqs. (64)-(65)] The step "use prime-average analogue of Oppermann" does not imply \bar\pi(\bar p_{n+1}^2)-\bar\pi(\bar p_n^2)>2\bar g_n. The Oppermann analogue proved in Sec. 5.4 gives, for each real m\ge 2, at least one averaged prime in (m^2,(m+1/2)^2) and in ((m-1/2)^2,m^2), but these local existence statements are not additive: a single averaged-prime value can lie in more than one such interval as m varies, so the number of distinct averaged primes in (\bar p_n^2,\bar p_{n+1}^2) is not bounded below by 2\bar g_n through this argument. Consequently Eq. (65) and the derivation of Eqs. (66)-(67) are unsupported. The Brocard analogue is likely recoverable from a density estimate using \bar g_k<\ln k, but the manuscript must supply that argument.
[Sec. 5.6, Eq. (70)] The second inequality in Eq. (70) has the wrong direction. Since \bar p_n < p_n/2 < (1/2)n\ln(n\ln n) = (1/2)n\ln n(1+\ln\ln n/\ln n), the quotient n\ln n/\bar p_n is larger than 2/(1+\ln\ln n/\ln n), not smaller. The displayed chain therefore cannot yield Q<2-\ln\bar p_n. The intended conclusion follows instead from the already-proved bound \bar g_n/\bar p_n<2/n (Eq. (36)), which gives n\bar g_n/\bar p_n<2. Please correct the derivation and re-check the small-n verification for n=1,...,11.

minor comments (4)

[Secs. 5.8 and 5.9, Eqs. (81) and (87)] The claimed analytic thresholds are numerically false: \ln((n/2)\ln(n/2))>2 first holds at n=10, not n=5, and \ln(p_n/(2\ln p_n)\ln(n/2))>2 first holds at n=11, not n=5. The final statements n\ge 6 and n\ge 7 appear to be supported by the explicit checks, but the incorrect thresholds should be removed or corrected.
[Sections 2-5, finite checks] The proofs repeatedly rely on unlisted finite computations phrased as "explicitly checking smaller integers", "smaller values of x", and "the first 55 average primes". These checks should be made reproducible by stating the exact ranges and either tabulating the results or providing the verification code.
[Sec. 5.4] The phrase "for integer n\ge 3 corresponding to m\ge\sqrt 5" is inaccurate: \bar\pi(m^2)\ge 3 first occurs when m^2\ge \bar p_3=10/3, i.e. m\ge\sqrt{10/3}. The subsequent argument is not affected, but the statement should be corrected.
[Introduction] There is a typo in the introduction: "we shall soon se" should be "we shall soon see".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the averaged-prime theorems are derived from external Rosser-Schoenfeld, Dusart, Mandl, and Hassani estimates, with no target conjecture assumed.

full rationale

The paper's central derivation is self-contained against external number-theoretic estimates. The key gap bound \bar g_n < \ln n for n\ge 3 (Eq. 29) follows from Rosser-Schoenfeld's bounds on p_n and \vartheta(x) (Eqs. 8-12), the integral lower bound on \bar p_n (Eq. 15), and explicit checking of small integers. The intended lower bound \bar g_n > \ln n/2 (Eq. 31) is built from Mandl's inequality \bar p_n < p_n/2 (Eq. 5) and Rosser's bound p_n > n\ln n (Eq. 8); it does not assume the conclusion. The averaged analogues in Section 5 are then deduced from these gap bounds together with Dusart's estimates (Eqs. 16-17) and, in the Nicholson/Farhadian cases, from \bar p_n > p_{[n/2]} (Eq. 7). No ordinary-prime conjecture is used as an input. The self-citations [13,14] appear only in the Discussion as numerical motivation for the ordinary conjectures and are not load-bearing. The paper explicitly acknowledges that the main formulation effort is in designing compelling averaged analogues (Sections 5 and 6), which is a design choice, not circularity. A skeptical reading points to an algebraic slip in Eq. (30) that leaves the lower-bound proof incomplete as written; that is a correctness concern, not a circularity, since a repair via Hassani's bound (Eq. 6) and finite checks would not assume the desired inequality. Overall, no step in the derivation chain reduces to its own input.

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

No free parameters or invented entities appear. The central claim rests entirely on cited explicit estimates and unlisted finite checks.

assumptions (6)

standard math Rosser: p_n > n ln n for all n≥1
Invoked in Sections 2, 4, and 5 for lower bounds on p_n and ¯p_n.
standard math Rosser-Schoenfeld: p_n < n ln(n ln n) for n≥6
Upper bound used to derive ¯g_n < ln n in Section 4.
standard math Rosser-Schoenfeld: |ϑ(x)-x| < x/(2 ln x) for x≥563, with small-case check for n≥26
Used in Section 2 lemma to prove g_n = p_{n+1}-p_n < n.
standard math Dusart: π(x) > x/(ln x-1) for x>5393 and π(x) < x/(ln x-139/125) for x>e^{139/125}
Used in Section 3 and Section 5.3 for counting averaged primes.
standard math Mandl and Hassani bounds: ¯p_n < p_n/2 for n≥9 and ¯p_n < p_n/2 - n/14 for n≥10
Used in Sections 4 and 5 to control ¯p_n from above.
domain assumption Unlisted finite computations are correct for n up to about 440 and for the first 55 averaged primes
The proofs of the small-n domains rely on 'explicitly checking smaller integers' without shipping a table or code.

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Cite this review

Pith. "Pith review of On the arithmetic average of the first $n$ primes." pith.science (2026). https://pith.science/paper/COY2HEYN

@misc{pith2026250504951,
  author       = {Pith},
  title        = {Pith review of: On the arithmetic average of the first $n$ primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/COY2HEYN}},
  note         = {Machine review of arXiv:2505.04951}
}

abstract

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \to \bar p_n$ is extremely easy to invert, $p_n = n\bar p_n - (n-1)\bar p_{n-1}$, it is clear that these two sequences $p_n \longleftrightarrow \bar p_n$ must ultimately carry exactly the same information. But the averaged sequence $\bar p_n$, while very closely correlated with the primes, ($\bar p_n \sim {1\over2} p_n$), is much "smoother'', and much better behaved. Using extensions of various standard results I shall demonstrate that the prime-averaged sequence $\bar p_n$ satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures, they are theorems.) The crucial key to enabling this pleasant behaviour is the "smoothing'' process inherent in averaging. Whereas the asymptotic behaviour of the two sequences is very closely correlated the local fluctuations are quite different.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$
math.NT 2025-07 reject novelty 5.0 of 10

Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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work page

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