Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$

Matt Visser (Victoria University of Wellington)

REVIEW 3 major objections 4 minor 19 references

Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the log-primorial sequence $\vartheta_n=\ln([p_n]\#)$ turns suitably formulated $\vartheta$-analogues of nine famous prime-gap conjectures into unconditional theorems, with the regularity of the gaps…

desk verdict Two concrete errors sink the abstract's claim that all nine ϑ-analogues are theorems; the gap bounds and some analogues are correct and worth keeping. read the letter →

arxiv 2507.14410 v1 pith:PP6ZSS5J submitted 2025-07-18 math.NT

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

classification math.NT MSC 11A4111N05

keywords log-primorialsequencefirstChebyshevfunctionϑ-gapsϑ-analoguesofprimeconjecturesCramérconjectureAndricaFiroozbakhtgaps

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 claims that replacing the $n$th prime $p_n$ by its cumulative logarithm, the log-primorial $\vartheta_n = \vartheta(p_n) = \sum_{i=1}^n \ln p_i = \ln([p_n]\#)$, turns suitably formulated $\vartheta$-analogues of nine famous prime-gap conjectures into theorems. Because the gaps are $\mathfrak{g}_n = \vartheta_{n+1}-\vartheta_n = \ln p_{n+1}$, their relative size $\mathfrak{g}_n/\vartheta_n$ is provably of order $1/n$, lying between $9/(10n)$ and $9/(5n)$ for all $n$. The paper proves $\vartheta$-analogues of the Cramér, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures unconditionally from explicit bounds on primes and on the first Chebyshev function. Since $p_n$ and $\vartheta_n$ encode the same information, the point is that the log-primorial ordering exhibits, as theorems, the regular behaviour that remains conjectural for the primes themselves.

What carries the argument

The mechanism is the $\vartheta$-gap $\mathfrak{g}_n = \vartheta_{n+1}-\vartheta_n = \ln p_{n+1}$. Because $\mathfrak{g}_n$ is the logarithm of the next prime rather than a difference of primes, its size relative to $\vartheta_n$ is provably tiny: the paper establishes $9/(10n) < \mathfrak{g}_n/\vartheta_n < 9/(5n)$ for all $n\ge 1$ and the equivalent comparison $(22/25)\ln p_n/p_n < \mathfrak{g}_n/\vartheta_n < 2\ln p_n/p_n$. A secondary mechanism is the counting function $\pi_\vartheta(x)$ and the constants $M^*=\sup_i \vartheta_i/p_i$ and $m^*=\inf_i \vartheta_i/p_i$; these convert the gap bounds into interval-counting lower and upper bounds that power the Legendre, Oppermann, and Brocard analogues.

What would settle it

Compute the $\vartheta$-count on the interval $(4,9]$: the values $\vartheta_4 = \ln 210 \approx 5.35$ and $\vartheta_5 = \ln 2310 \approx 7.75$ lie inside, so $\pi_\vartheta(9)-\pi_\vartheta(4)=2$, while the largest contained gap is $\ln 11 \approx 2.398$ and $(9-4)/\ln 11 \approx 2.085 > 2$, contradicting (4.9). The same check can be repeated on any short interval whose right endpoint is not a $\vartheta$-value; the inequality fails whenever the largest contained gap exceeds the average spacing.

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Extended reading notes

Core claim

The central discovery is that the log-primorial sequence $\vartheta_n = \ln([p_n]\#)$ has provably regular gaps: $\vartheta_{n+1}-\vartheta_n = \ln p_{n+1}$, so the relative gap lies in the narrow band $9/(10n) < \mathfrak{g}_n/\vartheta_n < 9/(5n)$, and also $(22/25)\ln p_n/p_n < \mathfrak{g}_n/\vartheta_n < 2\ln p_n/p_n$. From these gap bounds, the paper derives explicit upper and lower bounds on the counting function $\pi_\vartheta(x) = \#\{i : \vartheta_i \le x\}$, then uses them to prove $\vartheta$-analogues: $\vartheta$-Andrica holds with maximal step $\sqrt{\ln 6}-\sqrt{\ln 2}\approx 0.506 < 1$; $\vartheta$-Firoozbakht holds for $n\ge 4$; $\vartheta$-Fourges holds for all $n\ne 2$; $\vartheta$-Nicholson holds for $n\ge 3$; $\vartheta$-Farhadian holds for $n\ge 5$; and $\vartheta$-Legendre, $\vartheta$-Oppermann, and $\vartheta$-Brocard follow from the counting bound $\pi_\vartheta(x)-\pi_\vartheta(y) > (x-y)/(\ln x + 1.059660101)$. The paper stresses that this does not transfer to the ordinary prime conjectures, since the identity $p_n = \exp(\vartheta_n-\vartheta_{n-1})$ does not naively convert $\vartheta$-gap theorems into prime-gap theorems.

Load-bearing premise

The load-bearing premise is the interval-counting estimate (4.9), which asserts that the number of $\vartheta$-values in an interval $(y,x]$ exceeds $(x-y)$ divided by the largest $\vartheta$-gap contained in that interval; this premise is false, since the interval $(4,9]$ contains two $\vartheta$-values while $(9-4)/\ln 11 \approx 2.085$ already exceeds two.

Editorial extensions

If this is right

The $\vartheta$-analogue of the Cramér conjecture is a trivial identity, since $\vartheta_{n+1}-\vartheta_n=\ln p_{n+1}$ is far smaller than $(\ln p_n)^2$; the $\vartheta$-analogue of Andrica is proved in the strong form $\sqrt{\vartheta_{n+1}}-\sqrt{\vartheta_n} \le \sqrt{\ln 6}-\sqrt{\ln 2} \approx 0.506 < 1$ for all $n$.
If the counting bound (4.19) is valid, then $\vartheta$-Legendre holds with many $\vartheta$-values between consecutive squares, $\vartheta$-Oppermann holds between consecutive half-squares for $m\ge 1$ and $m\ge 2$, and $\vartheta$-Brocard holds with at least $2\mathfrak{g}_n$ $\vartheta$-values between $p_n^2$ and $p_{n+1}^2$ for all $n$.
$\vartheta$-Firoozbakht holds for $n\ge 4$, meaning $(\vartheta_n)^{1/n}$ is eventually decreasing; $\vartheta$-Fourges holds for all positive integers except $n=2$; $\vartheta$-Nicholson holds for $n\ge 3$; $\vartheta$-Farhadian holds for $n\ge 5$.
Because $p_n = \exp(\vartheta_n-\vartheta_{n-1})$, the $\vartheta_n$ sequence and the primes encode the same information, but the paper explicitly shows that the naive exponential conversion does not produce proofs of the ordinary Cramér, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, or Farhadian conjectures.

Reading between the lines

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

Editorial inference: the failure of (4.9) is not confined to the one example; any interval in which the largest contained $\vartheta$-gap exceeds the average spacing will falsify the lower bound, so the counting-based proofs of the Legendre, Oppermann, and Brocard analogues need a repaired estimate, for instance subtracting an endpoint gap or bounding the maximum gap by $\ln x - \ln y$ rather than
Editorial inference: the qualitative point that log-primorial gaps are smooth is independent of the counting inequality; a corrected counting lemma would likely preserve the spirit of the Legendre, Oppermann, and Brocard analogues even if the numerical ranges change.
Editorial inference: the same 'cumulative logarithm' construction could be applied to other increasing sequences of integers to turn hard gap conjectures into provable statements about the cumulative sums of their logarithms, but the identity $\mathfrak{g}_n=\ln p_{n+1}$ is what makes the comparison to the original prime conjectures exact here.

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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

Two concrete errors sink the abstract's claim that all nine ϑ-analogues are theorems. That's the headline.

The paper is not without merit. The relative-gap bounds (3.15) — 0.9/n < g_n/ϑ_n < 1.8/n — are new and correct, and they make the Cramer, Andrica, Firoozbakht, and Fourges analogues straightforward and rigorous. The author is also honest in Section 6: these results don't tell us much about ordinary primes. That's a fair admission, and I appreciate it.

But the errors are load-bearing. First, the ϑ-analogue of Nicholson, Eq. (5.36), is false. The proof needs ln(n[ln n − 1/2]) > 9/5, which holds only for n ≥ 6, not n ≥ 4. The 'explicit checking' of smaller integers fails: at n=3 the LHS is (ln210/ln30)^3 ≈ 3.88, while the RHS is 3(ln3 − 1/2) ≈ 1.80. So (5.36) is not a theorem.

Second, the counting lemma (4.9) asserts #{ϑ_i : y<ϑ_i≤x} > (x−y)/max-gap. That inequality is false: spacing alone gives you at least floor(L/G) − 1, not L/G. For (4,9] the count is two, but (9−4)/ln7 ≈ 2.57. This lemma feeds the Legendre, Oppermann, and Brocard analogues, so their proofs collapse. A corrected bound would carry an extra −1 and would be too weak for the claimed conclusions; those three analogues may still be provable by other means, but not as written.

The external references (Rosser–Schoenfeld, Dusart, Platt–Trudgian) are standard and used correctly. The self-citations are not a problem by themselves, though they are frequent. The paper reads as an honest attempt that went off the rails in two spots.

I'd tell a serious referee to check the small-n cases carefully and to ask for a repair of the counting argument. The idea has legs and the gap bounds are worth keeping. As it stands, the central claim is unsupported, so this is not publishable in current form. But it deserves referee time rather than a desk reject: the errors are specific, and the correct parts are clearly useful. My recommendation: send to review, with a note to verify the numerics and the counting lemma.

Referee Report

3 major / 4 minor

Summary. The paper studies the sequence ϑ_n = ϑ(p_n), the first Chebyshev function evaluated at the nth prime, equivalently the logarithm of the primorial. Using known explicit bounds on primes and on ϑ(x), together with elementary estimates for the gaps g_n = ϑ_{n+1} − ϑ_n = ln p_{n+1}, the author claims to prove ϑ-analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. The central assertion is that these analogues are theorems rather than conjectures. The paper also correctly notes in §6 that these results do not directly imply the ordinary prime conjectures.

Significance. If the claims were correct, the observation that the log-primorial sequence is substantially smoother than the prime sequence would be mildly interesting and would provide a clean set of unconditional inequalities. The paper makes appropriate use of classical external results and is honest about the lack of implications for the usual prime conjectures. However, the main claim is not supported: at least one of the stated analogues is numerically false, and a key counting lemma used for three other analogues is false. The small-case checks are asserted without presenting any data or code, so the many 'explicitly checking' statements are not independently verifiable from the manuscript. The paper therefore does not establish the claimed set of theorems.

major comments (3)

[§4, Eq. (4.9)] The counting lower bound in Eq. (4.9) is false. For the interval (4,9], there are exactly two ϑ-values, namely ϑ_4 = ln 210 ≈ 5.347 and ϑ_5 = ln 2310 ≈ 7.745. The maximum ϑ-gap completely contained in this interval is ϑ_5 − ϑ_4 = ln 11 ≈ 2.398, so the right-hand side of (4.9) is 5/ln 11 ≈ 2.085, which exceeds 2. The interval counting estimate therefore fails. The derived bounds (4.12) and (4.19) are also false: for (x,y) = (7.7, 3.5), the left side is 1, while (x−y)/(ln x + 1.059660101) ≈ 4.2/(ln 7.7 + 1.059660101) ≈ 1.354 > 1.
[§5.8, Eq. (5.36)] The asserted ϑ-analogue of Nicholson is false as stated. For n = 3, the left side (ϑ_4/ϑ_3)^3 = (ln 210/ln 30)^3 ≈ 3.884, while the right side 3(ln 3 − 1/2) ≈ 1.796, so the inequality fails. It also fails at n = 4: the left side is approximately 4.400, while the right side is 3.545. The preceding threshold condition ln(n[ln n − 1/2]) > 9/5 is itself incorrect: it holds for n ≥ 6, not for n ≥ 4, since for n = 5 the left side is about 1.713 and for n = 6 about 2.047.
[§5.3–5.5 and §5.8] Because the false counting bound (4.19) is the basis for the lower estimates in the Legendre analogue (5.8), the Oppermann analogue (5.9)–(5.10), and the Brocard analogue (5.14)–(5.15), the proofs of those three results are invalid as written. In addition, the logical step leading to (5.32) in §5.8 is problematic: from ϑ_n > n[ln n − 1/2] one cannot strengthen (5.31) to (5.32), since the new right-hand side is smaller than ϑ_n, not larger; the claimed inequality must be proved directly, and the direct small-case check contradicts it at n = 3 and n = 4.

minor comments (4)

[§6, Eq. (6.1)] Equation (6.1) is misnotated: it reads g_n = p_{n+1} − p_n = exp(g_n) − exp(g_{n−1}), conflating the ϑ-gap g_n = ln p_{n+1} with the ordinary prime gap; it should state p_{n+1} − p_n = exp(g_n) − exp(g_{n−1}).
[§5.8] The phrase 'noting that for n ≥ 1 we have ϑ_n > n[ln n − 1/2]' as justification for strengthening the inequality is misleading, since the smaller right-hand side makes the new inequality stricter, not weaker; this is a logical point that should be clarified.
[Throughout] The manuscript repeatedly relies on 'explicitly checking smaller integers' (for example, Eqs. (3.5), (3.10), (3.13), (5.5), (5.42)) but provides no tables, code, or reproducible data; the reader cannot verify these checks, and in at least one case (Eq. (5.36)) the check is actually wrong.
[§1] There is a typo in the introduction: 'Fahadian' should be 'Farhadian', and the names 'Cramer' and 'Fourges' are not consistently accented ('Cramér' and 'Fourgès').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ϑ-analogue theorems are derived from external classical prime estimates and prior published inequalities, not from the conjectures they purport to prove.

full rationale

The derivation chain is self-contained against external mathematics. The inputs are classical results (Rosser, Rosser–Schoenfeld, Dusart, Bertrand, Platt–Trudgian) and previously published parameter-free inequalities, including the author's own bounds on p_{n+1}, ϑ(x), and prime gaps; none of these inputs contains the target ϑ-analogue statements or is fitted to them. The counting bound (4.9) and the numerical claims about the Nicholson analogue are independent mathematical assertions that can be checked directly, so even if some are false that is a correctness defect, not a circularity. The self-citations [4], [13], [14], [15], and [17] are used as ordinary prior theorems with explicit inequalities, and the central claim rests on the inequalities themselves rather than on the authority of the citation; under the review rules such citations are real evidence and do not raise the circularity score. No step was found where a fitted parameter is renamed as a prediction, a target conjecture is used as an assumption, or an analogue is true by definition in a way that makes the derivation equivalent to its input.

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

No free parameters are fitted to data. The paper relies on standard classical bounds in analytic number theory, and on unshown finite computations for small n. The false inequality (4.9) is a derivation error, not an extra axiom.

assumptions (6)

standard math Rosser's theorem: p_n > n ln n for n ≥ 1.
Used in (2.1) and in various lower bounds for p_n and ϑ_n.
standard math Rosser-Schoenfeld bounds on the first Chebyshev function, including |ϑ(x) - x| < x/(2 ln x) for x ≥ 563.
Used to derive the asymmetric bounds (2.8)-(2.9) on ϑ_n/p_n and the constants in the gap estimates.
standard math Dusart's explicit bounds on p_n and ϑ_n, e.g., ϑ_n ≤ n(ln n + ln ln n - 1 + (ln ln n - 2)/ln n) for n ≥ 198.
Used throughout Section 2 and 3 to establish upper and lower bounds on ϑ_n and p_n.
standard math Bertrand's postulate: p_{n+1} < 2 p_n.
Used in (3.3) to bound the ϑ-gap ratio.
domain assumption Finite explicit computations for small n are correctly performed.
The paper repeatedly extends inequalities from n ≥ N to n ≥ 1 by saying 'checking smaller integers by explicit computation' without showing the computations. If any of these finite checks were wrong, the validity domains of the bounds would change.
standard math Platt-Trudgian bound: M* < 1 + 7.5 × 10^{-7}.
Used in (4.4) and (4.18) to control the upper constant M* in the counting function section.

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Cite this review

Pith. "Pith review of Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$." pith.science (2026). https://pith.science/paper/PP6ZSS5J

@misc{pith2026250714410,
  author       = {Pith},
  title        = {Pith review of: Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PP6ZSS5J}},
  note         = {Machine review of arXiv:2507.14410}
}

abstract

The well-known sequence $\vartheta_n = \vartheta(p_n) = \sum_{i=1}^n \ln p_i= \ln\left([p_n]\#\right)$ exhibits numerous extremely interesting properties. Since $p_n = \exp(\vartheta_n - \vartheta_{n-1})$, it is immediately clear that the two sequences $p_n \longleftrightarrow \vartheta_n$ must ultimately encode exactly the same information. But the sequence $\vartheta_n$, while being extremely closely correlated with the primes, (in fact, $\vartheta_n \sim p_n$), is very much better behaved than the primes themselves. Using numerous suitable extensions of various reasonably standard results, I shall demonstrate that the sequence $\vartheta_n$ satisfies suitably defined $\vartheta$-analogues of the usual Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. (So these $\vartheta$-analogues are not conjectures, they are instead theorems.) The crucial key to enabling this pleasant behaviour is the regularity (and relative smallness) of the $\theta$-gaps $\mathfrak{g}_n = \vartheta_{n+1}-\vartheta_n= \ln p_{n+1}$. While superficially these results bear close resemblance to some recently derived results for the averaged primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, both the broad outline and the technical details of the arguments given and proofs presented are quite radically distinct.

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] [1]

Autour de la fonction qui compte le nombre de nombres premiers

Pierre Dusart, “Autour de la fonction qui compte le nombre de nombres premiers”, PhD Dissertation, Universite de Limoges, Limonges, France, 1998

work page 1998

[2] [2]

Bornes effectives pour certaines fonctions concernant les nombres premiers

Jean-Pierre Massias and Guy Robin, “Bornes effectives pour certaines fonctions concernant les nombres premiers”, Journal de Th´ eorie des Nombres de Bordeaux,8 #1 (1996), 215–242. https://www.jstor.org/stable/43972471

work page arXiv 1996

[3] [3]

The kth prime is greater than k(ln k + ln lnk − 1) for k ≥ 2

Pierre Dusart, “The kth prime is greater than k(ln k + ln lnk − 1) for k ≥ 2”, Mathematics of Computation. 68 (225) (1999) 411–415. doi:10.1090/S0025-5718-99-01037-6

work page doi:10.1090/s0025-5718-99-01037-6 1999

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On the arithmetic average of the first $n$ primes

Matt Visser, “On the arithmetic average of the first n primes”, Mathematics 13 (2025) 2279. doi:10.3390/math13142279 [arXiv: 2505.04951 [math.NT]]

work page Pith review arXiv 2025

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The n-th Prime is Greater than n log n

J. Barkley Rosser, “The n-th Prime is Greater than n log n”. Proceedings of the London Mathematical Society 45 (1939) 21-44, doi:10.1112/plms/s2-45.1.21

work page doi:10.1112/plms/s2-45.1.21 1939

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J. Barkley Rosser and Lowell Schoenfeld, “Approximate formulas for some functions of prime numbers”, Illinois J. Math. 6 #1 (March 1962) 64–94 doi:10.1215/ijm/1255631807

arXiv 1962

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Pierre Dusart, “Explicit estimates of some functions over primes”, Ramanujan J 45 (2018) 227–251. doi:10.1007/s11139-016-9839-4

doi:10.1007/s11139-016-9839-4 2018

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Pierre Dusart, “Estimates of Some Functions Over Primes without R.H.”, arXiv: 1002.0442 math.NT

arXiv