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The n-th prime exponentially

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper converts any effective error bound on the prime counting function into fully explicit, exponentially tight two-sided bounds on the nth prime.

desk verdict A repairable but currently false main theorem: the announced validity ranges ignore a missing denominator sign condition. read the letter →

arxiv 2504.14458 v1 pith:RD2AWSOK submitted 2025-04-20 math.NT

Matt Visser (Victoria University of Wellington) This is my paper · ORCID
classification math.NT MSC 11N0511N37
keywords nthprimecountingfunctionlogarithmicintegraleffectiveboundsexplicitestimatesnumbertheoremexponentialerrorgaps
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

The paper shows that from any known numerical error bound of the shape $|\pi(x) - \mathrm{Li}(x)| < a\,x\,(\ln x)^b \exp(-c\sqrt{\ln x})$, valid for $x \geq x_0$, one can derive two-sided bounds on $\pi(x)$ and on the $n$th prime $p_n$ whose error decays exponentially in $\sqrt{\ln n}$. Concretely, for $x$ above a computable threshold the prime count satisfies $\mathrm{Li}(x)/(1 + a f(x)) < \pi(x) < \mathrm{Li}(x)/(1 - a f(x))$, and for $n$ above a bounded threshold $n_*$ the $n$th prime satisfies $\mathrm{Li}^{-1}(n[1 - a f(n\ln n)]) < p_n < \mathrm{Li}^{-1}(n[1 + a f(n\ln n)])$, where $f(t) = (\ln t)^{b+1} \exp(-c\sqrt{\ln t})$. The same machinery converts every row in several published effective bounds, with explicit constants, into numerical brackets on individual primes, some valid already from $n = 1$. A sympathetic reader would care because this turns decades of hard effective estimates on prime distribution into directly usable statements about where actual primes sit, without invoking any unproved hypothesis.

What carries the argument

The load-bearing object is the function $f(t) = (\ln t)^{b+1} \exp(-c\sqrt{\ln t})$, the residual after pulling one factor of $x/\ln x$ out of the input error term. The key identity is $a\,x\,(\ln x)^b \exp(-c\sqrt{\ln x}) = a\,(x/\ln x)\,f(x)$, and the elementary inequality $x/\ln x < \pi(x)$ for $x > 17$ turns the absolute error bound into the relative error bound $|\pi(x)-\mathrm{Li}(x)| < a\,\pi(x)\,f(x)$, which is then rearranged into the $\mathrm{Li}/(1\pm a f)$ sandwich. Everything else is domain bookkeeping: $f$ has a single peak at $x_{\mathrm{peak}} = \exp([2(b+1)/c]^2)$, the classical bounds $p_n > n\ln n$ and $p_n < (1+e^{-1})n\ln n$ move the argument from $p_n$ to $\pi(x)\ln\pi(x)$, and an explicit interval bound guaranteeing $p_{n+1} < (4/3)p_n$ keeps the monotonicity argument from overshooting the peak. The special case $b=-1$ makes $x_{\mathrm{peak}}=1$, eliminating the peak complication and giving the simplest final formulas.

What would settle it

Check a single advertised row: take its $(a,b,c,x_0)$, form $n_* \leq \max\{\pi(x_0),7,\pi((1+e^{-1})\exp([2(b+1)/c]^2))\}$, and compute the first $n \geq n_*$ at which $1 - a\,(\ln[n\ln n])^{b+1}\exp(-c\sqrt{\ln[n\ln n]}) \leq 0$; if such an $n$ exists, the lower $p_n$ interval is not valid as stated for that row. A direct numerical check of $|\pi(p_n)-n| < a\,n\,(\ln[n\ln n])^{b+1}\exp(-c\sqrt{\ln[n\ln n]})$ over a long initial range would also expose any error in the derivation.

Watch

Extended reading notes

Core claim

The central claim is a conversion theorem. Start with any fully explicit bound $|\pi(x) - \mathrm{Li}(x)| < a\,x\,(\ln x)^b \exp(-c\sqrt{\ln x})$ for $x \geq x_0$, with $b \geq -1$ and $c > 0$. Because $x/\ln x < \pi(x)$ for $x > 17$, the error can be rewritten as a relative error, $|\pi(x) - \mathrm{Li}(x)| < a\,\pi(x)\,f(x)$ with $f(x) = (\ln x)^{b+1} \exp(-c\sqrt{\ln x})$, and this rearranges to the sandwich $\mathrm{Li}(x)/(1 + a f(x)) < \pi(x) < \mathrm{Li}(x)/(1 - a f(x))$. Replacing $x$ by $p_n$, using the classical bounds $p_n > n\ln n$ and $p_n < (1+e^{-1})n\ln n$ to move the argument of $f$ from $p_n$ to $n\ln n$, yields $\mathrm{Li}^{-1}(n[1 - a f(n\ln n)]) < p_n < \mathrm{Li}^{-1}(n[1 + a f(n\ln n)])$ for all $n \geq n_*$, where $n_* \leq \max\{\pi(x_0), 7, \pi((1+e^{-1})\exp([2(b+1)/c]^2))\}$. The error term is $\exp(-c\sqrt{\ln[n\ln n]})$, so it vanishes faster than any power of $\ln n$; the paper stresses that all constants and thresholds are explicit, that none of the bounds are claimed optimal, and that the same conversion works for any future bound of the same shape.

Load-bearing premise

The whole construction takes as input the bound $|\pi(x)-\mathrm{Li}(x)| < a\,x\,(\ln x)^b\exp(-c\sqrt{\ln x})$ with the stated constants, and for the lower $p_n$ interval it additionally requires $1 - a\,(\ln[n\ln n])^{b+1}\exp(-c\sqrt{\ln[n\ln n]}) > 0$ across $n \geq n_*$, which the advertised threshold does not by itself guarantee for the large-$a$ rows.

Editorial extensions

If this is right

  • For every input bound in Tables I–III, the method yields explicit numerical brackets on $p_n$ with stated thresholds; several rows are valid from $n_*=1$, so the bounds apply immediately rather than only asymptotically.
  • Because the error decays as $\exp(-c\sqrt{\ln[n\ln n]})$, these intervals eventually become narrower than any estimate whose error is a fixed power of $\ln n$, including the standard asymptotic expansions for $p_n$.
  • The special case $b=-1$ gives $p_n < \mathrm{Li}^{-1}(n[1 + a\exp(-c\sqrt{\ln(n\ln n)})])$ and the matching lower bound, with threshold $n_* \leq \max\{\pi(x_0),7\}$ and no peak-related adjustments.
  • Reading the same inequality at $p_n$ and just below $p_{n+1}$ and using the triangle inequality produces an explicit prime-gap bound $p_{n+1} < \mathrm{Li}^{-1}(\mathrm{Li}(p_n) + 2a\,n\,f(n\ln n))$, valid for $n \geq n_*$.
  • Any future improvement in the constants $(a,b,c,x_0)$ for $|\pi(x)-\mathrm{Li}(x)|$ flows through the same formulas to sharper explicit brackets on individual prime locations.

Reading between the lines

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

  • The derivation is generic in the input bound, so any future or conditional estimate of the same exponential shape, whether unconditional or assuming unproved hypotheses, would immediately produce the corresponding explicit $p_n$ brackets without new work.
  • The advertised $n_*$ is a guaranteed threshold, not an optimal one; for the large-$a$ rows the lower bracket $1 - a\,(\ln[n\ln n])^{b+1}\exp(-c\sqrt{\ln[n\ln n]})$ can turn non-positive before $n$ reaches $n_*$, so a user should verify positivity at the claimed threshold before applying the lower interval.
  • The $\mathrm{Li}/(1\pm a f)$ sandwich could be iterated to produce a deterministic algorithm that locates $p_n$ without sieving up to $p_n$, with running cost governed by the exponential decay of $f$.
  • The same machinery should transfer to bounds on the weighted prime-counting function $\vartheta(x)$, where the $b=0$ relaxation is the natural fit, yielding exponentially tight explicit estimates for that function too.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes a method for converting known effective bounds of the form |π(x) − Li(x)| < a x (ln x)^b exp(−c√ln x) into two-sided exponentially tight bounds on π(x) and on the nth prime p_n. The main results are displayed in Eqs. (3.7) and (4.16), with explicit validity ranges x_* ≤ max{x0,17} and n_* bounded by π(x0), π(17), and π((1+e^{-1}) exp([2(b+1)/c]^2)). Numerical tables apply the method to bounds of Trudgian, Fiori–Kadiri–Swidinsky, and Johnston–Yang, and a special case b = −1 is treated in Section 5.

Significance. If the stated ranges were correct, the paper would offer a simple, fully explicit mechanism for converting recent effective prime-number-theorem bounds into bounds on p_n that decay exponentially in √log n, complementing the Cipolla-type asymptotic expansions. The algebraic idea—replacing x by π(x) and rearranging the input bound—is attractive, and the paper does not fit free parameters to data; the input bounds are cited theorems and the manipulations are self-contained. However, the central inequalities are not valid on the advertised ranges, so the paper's main claims as written are not established. The strengths of the approach (no fitted parameters, fully explicit proof structure) are real, but they do not overcome the load-bearing errors described below.

major comments (3)
  1. [3, Eq. (3.7) and Abstract] The displayed upper bound π(x) < Li(x)/(1 − a f(x)) is obtained from the right inequality π(x) − Li(x) < a π(x) f(x) by dividing by 1 − a f(x). This division is only valid when 1 − a f(x) > 0; if the denominator is negative, the inequality direction reverses and the displayed upper bound is false. The paper neither states nor verifies this positivity condition. For the Fiori–Kadiri–Swidinsky bound in Table I (a = 9.2211, b = 1/2, c = 0.8476), at x = 17 one has a f(17) ≈ 10.6, so 1 − a f(17) is about −9.6; the asserted upper bound then gives π(17) < Li(17)/(−9.6) < 0, contradicting π(17) = 7. The same failure occurs for the Johnston–Yang bound and in the special case of Section 5. Consequently the abstract's claim x_* ≤ max{x0,17} is false, and Eq. (3.7) is not a valid theorem as stated.
  2. [4, Eq. (4.16) and Table IV] The lower bound p_n > Li^{-1}(n[1 − a f(n ln n)]) is only a meaningful two-sided localization when 1 − a f(n ln n) > 0; otherwise the right-hand side is negative (or undefined at n = 1), so the inequality is trivially true and provides no genuine lower bound on p_n. The stated n_* bound does not ensure this positivity. For the FKS parameters, Table IV lists n_* = 1, and for n = 7 one already has a f(n ln n) > 1, so the advertised effective two-sided bounds on p_n are not established on the claimed ranges. The large-a rows of Table V suffer from the same problem in the special case b = −1.
  3. [4, Eqs. (4.10)–(4.12)] The proof that π(x) ln π(x) > x_peak is 'certainly satisfied for x > (1 + e^{-1}) x_peak' is logically invalid: the fact that x exceeds a threshold does not imply that the largest prime p_{π(x)} ≤ x also exceeds that threshold. Using only Nagura's bound p_{n+1} < (4/3)p_n, one can guarantee a prime above (3/4)x, which may still be below (1+e^{-1})x_peak. Thus the derivation of (4.12) has a genuine gap, and hence the subsequent n_* bound in (4.15) and in the abstract is not proven by the given argument.
minor comments (3)
  1. [Throughout] The paper repeatedly writes 'Cippola' for the correct spelling 'Cipolla' (Sections 4 and 7), and there are typographical issues such as 'T able' in the abstract; a careful proofreading is needed.
  2. [Table IV] Table IV is difficult to read because the numerical entries are packed without clear separators; please format the columns so that the values of x_*, n_0, n_peak, and n_* are unambiguous, especially in rows such as '9.2211 1/2 0.8476 2 275 789 2 1 24 104 1'.
  3. [4, Table IV] The procedure used to 'explicitly evaluate' the entries labelled x_* and n_* is not described; please state precisely what was checked, over what range, and ideally provide reproducible code or a verification algorithm.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a self-contained algebraic consequence of externally cited bounds on |π(x) − Li(x)|.

full rationale

The paper's central claims are conditional transformations of known external estimates, not restatements of its own assumptions. The input (1.1), |π(x) − Li(x)| < a x (ln x)^b exp(−c√ln x), is cited from Trudgian, Fiori–Kadiri–Swindisky, and Johnston–Yang — all independent published results. The paper adds no fitted parameters and does not tune a, b, c, or x0 to data. The derivation proceeds by the elementary inequality x/ln x < π(x) for x > 17 (equation 3.3), the definition f(x) = (ln x)^{b+1} exp(−c√ln x) (equation 3.2), and algebraic rearrangement to (3.7), then uses standard bounds p_n > n ln n and p_n < n ln(n ln n) to localize the argument of f at p_n (equations 4.8–4.16). None of these steps defines a conclusion in terms of itself. The self-citations [4] and [10] are to the author's previous work, but they are not load-bearing: [4] is invoked only as a heuristic variant for relaxing bounds in Section 2, and [10] is mentioned only as related work in Section 6. The missing positivity condition 1 − a f(x) > 0 in equation (3.7) is a substantive mathematical correctness issue — as the skeptical analysis shows, the displayed upper bound can fail for some Table I inputs at small x — but it is not a circularity. A false or unproven inequality is not the same as a prediction that reduces to its input by construction. Therefore, under the circularity-specific criteria, the appropriate finding is no significant circularity, score 0.

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

The paper introduces no new fitted free parameters; the constants a,b,c,x0 are inherited from the cited input bounds. It relies on several well-known theorems about primes and on the monotonicity of Li. No new physical or mathematical entities are postulated.

assumptions (6)
  • domain assumption Rosser's theorem: p_n > n ln n for n ≥ 1
    Cited from Rosser (1939), used in eq (4.6) to move from f(x) to f(π(x) ln π(x)).
  • domain assumption Nagura's bound: p_{n+1} < (4/3) p_n for n ≥ 5
    Cited from Nagura (1952), used in eq (4.4)-(4.5) to establish x > p_{π(xpeak)+1} for x > (4/3)xpeak.
  • domain assumption Rosser's upper bound: p_n < n ln(n ln n) for n ≥ 6
    Cited from Rosser (1941), used in eq (4.8) to derive π(x) ln π(x) > p_{π(x)}/(1+e⁻¹).
  • domain assumption Input bound |π(x)−Li(x)| < a x (ln x)^b exp(−c sqrt(ln x)) for x ≥ x0
    Treated as an external theorem, Table I lists the specific instances from Trudgian, FKS, and JY. The paper's proof is conditional on this.
  • standard math x/ln x < π(x) for x > 17
    Used in eq (3.3) to replace x/ln x by π(x) in the error bound, yielding |π−Li| < a π(x) f(x).
  • standard math Li(x) is strictly increasing on [2,∞), so Li⁻¹ is monotone
    Used to invert the error estimate at x = p_n into bounds on p_n via Li⁻¹.

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Cite this review

Pith. "Pith review of The n-th prime exponentially." pith.science (2026). https://pith.science/paper/RD2AWSOK

@misc{pith2026250414458,
  author       = {Pith},
  title        = {Pith review of: The n-th prime exponentially},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RD2AWSOK}},
  note         = {Machine review of arXiv:2504.14458}
}
abstract

From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem: For $x \geq x_*$ where $x_*\leq \max\{x_0,17\}$ we have: \[ {\mathrm{Li} \over 1+a\; (\ln x)^{b+1} \; \exp\left(-c\; \sqrt{\ln x}\right)} < \pi(x) < {\mathrm{Li} \over 1-a \;(\ln x)^{b+1} \; \exp\left(-c\; \sqrt{\ln x}\right)}. \] Furthermore, it is possible to establish exponentially tight effective upper and lower bounds on the location of the $n^{th}$ prime. Specifically: \[ p_n < \mathrm{Li}^{-1} \left( n \left[1+ a \;(\ln[n\ln n])^{b+1} \; \exp\left(-{c}\; \sqrt{\ln[n\ln n]}\right)\right] \right); \qquad (n\geq n_*). \] \[ p_n > \mathrm{Li}^{-1} \left( n \left[1- a \;(\ln[n\ln n])^{b+1} \; \exp\left(-{c}\; \sqrt{\ln[n\ln n]}\right)\right] \right); \qquad (n\geq n_*). \] Here the range of validity is explicitly bounded by some $n_*$ satisfying \[ n_* \leq \max\left\{\pi(x_0),\pi(17), \pi\left( (1+e^{-1}) \exp\left( \left[2(b+1)\over c\right]^2\right)\right) \right\}. \] Many other fully explicit bounds along these lines can easily be developed.

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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

  1. 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

10 extracted references · 5 canonical work pages · cited by 1 Pith paper

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