Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals

Glenn Bruda

REVIEW 1 major objections 6 minor 12 references

Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals

T0 review · 1 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For every positive integer k, the reciprocal of the k-th Hardy-Littlewood logarithmic integral has an explicit full asymptotic expansion in powers of 1/log x, with coefficients given by a recurrence that generalizes the indecomposable…

desk verdict A correct, modest generalization of Panaitopol's reciprocal li expansion; the proof has a repairable rigor gap and a sign typo, but the main result stands. read the letter →

arxiv 2412.19866 v1 pith:2HTN6L2G submitted 2024-12-27 math.CO math.NT

Glenn Bruda This is my paper · ORCID

classification math.COmath.NT MSC 05A1511N0541A6005A05

keywords asymptoticexpansionsHardy-Littlewoodlogarithmicintegralsreciprocalintegralindecomposablepermutationsformalpowerseriesprimecountingfunctionconjectureexponential

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 defines a family of integer sequences by a recurrence and uses them to write down a full asymptotic expansion, for each positive integer k, of the reciprocal of the k-th Hardy-Littlewood logarithmic integral $\mathrm{li}_k(x)$. These integrals are the ones appearing in the first Hardy-Littlewood conjecture on prime k-tuples, so the expansion supplies higher-order corrections beyond the leading term. In the case k=1 the result reproduces the known expansion for the reciprocal of the prime counting function, since $\pi(x)$ and $\mathrm{li}(x)$ have the same asymptotic behavior. A sympathetic reader would care because the proof ties a combinatorial counting problem, the enumeration of indecomposable permutations and its generalizations, to analytic number theory.

What carries the argument

The load-bearing object is the coefficient sequence $a_n^{(k)}$ together with the generating-function identity of Proposition 2.2: $$1 - (k-1)!\left(\sum_{n\ge 0}(n+k-1)!\,x^n\right)^{-1} = kx + \sum_{n\ge 1} $a_n^{{(k)}}$ $x^{{n+1}}$$$ as formal power series. This identity connects the recurrence to the reciprocal of a factorial series; when $x$ is replaced by $1/\log x$ and the factorial series is read through the asymptotic expansion of the exponential integral, it becomes the correction series in the expansion of $1/\mathrm{li}_k(x)$. The sequence's $k=1$ case counts indecomposable permutations, and the paper shows the higher-$k$ sequences play the same combinatorial role for the higher logarithmic integrals.

What would settle it

Evaluate $1/\mathrm{li}_2(x)$ numerically to high precision at $x=10^j$ for a fixed truncation order $N$ and check whether $\frac{(\log x)^{N+2}}{x}\left(1/\mathrm{li}_2(x) - \text{partial sum}_N(x)\right)$ stays bounded as $j$ grows. If it does not, the asserted asymptotic expansion for $k=2$ is false; checking the first coefficients against Proposition 2.2 would catch algebraic errors.

Watch

Extended reading notes

Core claim

The main result, Theorem 2.7, states that for each $k \in \mathbb{Z}^+$, as $x \to \infty$, $$\frac{1}{\mathrm{li}_k(x)} \sim \frac{(\log x)^k}{x}\left(1 - \frac{k}{\log x} - \sum_{n\ge 1}\frac{$a_n^{{(k)}}$}{(\log x)^{n+1}}\right),$$ where the coefficients $a_n^{(k)}$ are fixed by the recurrence $$(k-1)!\,$a_n^{{(k)}}$ = (n+k)! - k(n+k-1)! - \sum_{m=0}^{n-2} (n-m+k-2)!\,a_{m+1}^{(k)},$$ with $a_0^{(k)}=1$. The proof starts from a formal power-series identity for the generating function of these coefficients and transfers it into an asymptotic statement via the standard asymptotic expansion of the exponential integral and a change of variables. When $k=1$, the coefficients count indecomposable permutations and the expansion reduces to the known reciprocal prime-counting expansion.

Load-bearing premise

The load-bearing assumption is that substituting the divergent series for the exponential integral by its asymptotic expansion, and then reciprocating and re-expanding termwise, preserves the final asymptotic expansion; this operation is used in the proof of Theorem 2.7 without being stated or proved.

Editorial extensions

If this is right

For every $k$, the reciprocal $1/\mathrm{li}_k(x)$ now has a complete, explicitly computable asymptotic series in powers of $1/\log x$, with integer coefficients fixed by an elementary recurrence.
Setting $k=1$ recovers the known expansion for the reciprocal prime counting function, since $\mathrm{li}(x)-\pi(x)$ is much smaller than any term in the asymptotic sequence.
The coefficients $a_n^{(k)}$ generalize the indecomposable permutation counts, giving sequences originally studied combinatorially a role in the analytic theory of prime distributions.
The expansion supplies the correction terms needed if the first Hardy-Littlewood conjecture is strengthened from a leading-order asymptotic to a full asymptotic expansion for prime $k$-tuple counts.

Reading between the lines

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

If the first Hardy-Littlewood conjecture is ever promoted to a full asymptotic expansion for prime $k$-tuple counts, the coefficients $a_n^{(k+1)}$, up to the singular-series constant of the tuple, are the natural correction coefficients; this is an extrapolation, since the paper only says the expansion is 'salient' to the conjecture.
The $k=1$ equivalence between $1/\mathrm{li}(x)$ and $1/\pi(x)$ uses an error bound much larger than the individual terms of the new series, so comparing the expansion with numerical values of $\mathrm{li}_k(x)$ directly would be a sharper test than comparing with prime counts.
The recurrence likely admits a combinatorial interpretation for every $k$, since its $k=1$ case counts indecomposable permutations; such an interpretation would give an independent, purely finite check of the analytic coefficients.

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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

Quick take: this is a small but legitimate result. The paper generalizes Comtet's generating-function identity to the series with factorial coefficients shifted by k, and uses it to write down the full asymptotic expansion of 1/li_k(x). For k>1 that expansion is new, and the k=2 case checks out against direct integration by parts. The combinatorial connection to indecomposable permutations only really bites at k=1, but the generalization is natural.

What the paper does well: the formal step in Proposition 2.2 is clean and self-contained, and Theorem 2.7 is clearly stated. The coefficients a_n^(k) come from an independent recurrence, not from fitting the target integral, so there is no circularity. The relation to the first Hardy-Littlewood conjecture is stated honestly as auxiliary information, not as a stronger theorem.

The soft spots are minor but real. The proof of Theorem 2.7 substitutes the divergent formal series A(1/x) with its Poincaré asymptotic expansion x e^{-x} Ei(x) and then manipulates the resulting asymptotic series, including taking reciprocals, without citing or proving the standard theorem that rational operations preserve Poincaré expansions when the denominator's leading term is nonzero. That is exactly the step the stress-test note flags. It is a rigor gap rather than a counterexample: the expansion is correct, and the k=2 check confirms it. The fix is a one-line citation or a short argument.

Definition 2.5 also has a sign typo: the minus sign makes li_1(x) equal -li(x), contradicting the text and Lemma 2.6. This is clearly a typo, but it needs fixing.

A minor clarity point: the paper could state explicitly that Theorem 2.7 is the reciprocal of the standard asymptotic expansion of li_k. That would make the relationship to existing work plainer.

Who should read this: people who work with li_k in asymptotics or in computations for prime k-tuples. It is not a breakthrough, but it is a solid technical note. If it is submitted, I would send it to a referee; with the manipulation step cited and the typo corrected, it is acceptable. I wouldn't cite it in my own work unless I needed that specific expansion, but it deserves a place in the literature.

Referee Report

1 major / 6 minor

Summary. This paper defines, for every positive integer k, an integer sequence (a_n^{(k)}) by the recurrence in Definition 2.1 (with a_0^{(k)} = 1) and proves Proposition 2.2, a formal power series identity generalizing Comtet's generating function for indecomposable permutations: 1 − (k−1)! (∑_{n≥0} (n+k−1)! x^n)^{-1} = kx + ∑_{n≥1} a_n^{(k)} x^{n+1}. Proposition 2.4 (the standard asymptotic expansion of the exponential integral) and Lemma 2.6 (a reduction of li_k to li) are then combined with this identity to prove Theorem 2.7, which asserts the complete asymptotic expansion 1/li_k(x) ∼ (log x)^k/x (1 − k/log x − ∑_{n≥1} a_n^{(k)}/(log x)^{n+1}) for the reciprocal of the k-th Hardy-Littlewood logarithmic integral. For k = 1 the expansion reduces to the known expansion of 1/li(x), and through the remark following Proposition 1.2 to Panaitopol's expansion for 1/π(x). The Hardy-Littlewood application is presented as motivation, with an explicit caveat that the result only provides auxiliary information for a possible strengthening of Conjecture 1.3.

Significance. If the proof is completed as suggested in Major Comment 1, the main theorem is a correct, parameter-free, complete asymptotic expansion whose coefficients come from an independently defined combinatorial recurrence; there is no circularity and no fitted constant. The k = 2 case matches the direct integration-by-parts expansion, the k = 1 case reproduces the Comtet/Panaitopol system, and the coefficient sequences are identified with OEIS entries; these are genuine strengths that make the expansion a concrete, checkable target for any future strengthening of the first Hardy-Littlewood conjecture. The significance of the result is moderate: it is a clean unification of a classical combinatorial generating function with the asymptotic theory of logarithmic integrals, and the paper's claims about the conjecture are appropriately hedged. The principal weakness is the unstated asymptotic-manipulation step in the proof of Theorem 2.7, which is the load-bearing point of the paper and needs a short lemma or a precise citation.

major comments (1)

[Theorem 2.7, proof (step after 'Proposition 2.4 gives the relation A(1/x) ∼ xe^{−x}Ei(x); thus, ...')] This step substitutes the divergent formal series A(1/x) = ∑_{n≥0} n!x^{−n} by the function xe^{−x}Ei(x) inside the rational expression (k−1)!x^{1−k}/(A(1/x) − ∑_{i=0}^{k−2} i!x^{−i}) and then takes a reciprocal, without any stated justification. As written, the line is a formal manipulation of a divergent series, not a consequence of Definition 1.1. What is needed is the standard theorem that Poincaré asymptotic expansions are preserved under rational combinations when the denominator's leading asymptotic term is nonzero — here the denominator has leading term (k−1)!/x^{k−1} — and this theorem is neither stated, nor proved, nor cited. This step is load-bearing, because it is what turns the formal identity of Proposition 2.2 into the asymptotic statement. I verified the conclusion independently: for k = 2 the theorem's expansion reproduces 1/li_2(x) ∼ (log x)^2/x (1 − 2/log x − 2/(log x)^2 − 8/(log x)^3 − 44/(log x)^4 − ···) from the direct expansion of li_2, so the gap is a rigor gap rather than an error. The fix is local: add a lemma (if f(x) ∼ ∑ c_n x^{−n} with c_0 ≠ 0, then 1/f(x) admits the formal reciprocal expansion, and rational expressions inherit expansions by substitution), with a short proof or a precise citation, and verify its hypotheses for the specific denominator used.

minor comments (6)

[Proposition 2.2, proof] The first displayed line reads '(k−1)!I_k(x) = k!x − ∑_{n≥1}(k−1)!a_n^{(k)}x^{n+1}', but since I_k(x) = kx + ∑_{n≥1}a_n^{(k)}x^{n+1} the sign before the sum should be '+'; the subsequent line is consistent with the corrected sign, so this is a typo, but it is very confusing.
[Theorem 2.7, statement] The asymptotic sequence with respect to which the expansion holds is implicit; it should be stated explicitly (e.g., {x^{−1}(log x)^{k−m}}_{m≥0}) so that the statement conforms to Definition 1.1.
[Theorem 2.7, proof] The notation 'A(1/x) ∼ xe^{−x}Ei(x)' is an abuse of notation, since A(1/x) diverges for every finite nonzero x; it would be clearer to write that xe^{−x}Ei(x) has the asymptotic expansion ∑_{n≥0} n!x^{−n}.
[Introduction, remark after Proposition 1.2] The sentence 'Proposition 1.2 may be quickly obtained ... by following the k = 1 case of the proof of Theorem 2.7' refers forward to a theorem proved later; consider rephrasing, for example, 'by following the proof of Theorem 2.7 in Section 2 with k = 1'.
[References / novelty] Please confirm that the k ≥ 2 cases of Theorem 2.7 do not already appear in [5] or in the literature following [11]; if they do, the abstract's novelty claim should be adjusted accordingly.
[General] There are several typographical artifacts (e.g., 'C onsequently' in the abstract, 's eries' in the first paragraph of Section 1); a careful proofread is advised.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coefficient sequence is independently defined by a recurrence, and the proof derives the asymptotic expansion from the standard DLMF expansion of Ei, with no fitted parameter or self-citation chain.

full rationale

The paper's central claim, Theorem 2.7, gives an asymptotic expansion for 1/li_k(x) whose coefficients a_n^(k) are defined by the independent recurrence in Definition 2.1, involving only factorials and earlier coefficients, with no reference to li_k(x), Ei(x), or the target asymptotic expansion. Proposition 2.2 proves a formal power series identity relating these coefficients to the reciprocal of the series sum (n+k-1)! x^n. The proof of Theorem 2.7 then combines this formal identity with the standard, externally cited asymptotic expansion Ei(x) ~ e^x/x sum n!/x^n (Proposition 2.4, from DLMF) and the elementary integration-by-parts identity of Lemma 2.6. The coefficients are not fitted to any data from the logarithmic integrals, and no parameter is renamed as a prediction. The only notable gap is that the proof substitutes the divergent formal series A(1/x) with its Poincaré asymptotic expansion and then manipulates reciprocals without explicitly stating the standard theorem on asymptotic expansions of reciprocals; this is a rigor gap, not circularity. There are no self-citations at all, and no load-bearing appeal to the authors' prior work. The result is therefore self-contained relative to its stated external inputs, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof uses standard results (asymptotic expansion of Ei, formal power series) plus one unstated domain assumption: asymptotic power series are closed under composition and reciprocation. No free parameters or invented entities are introduced.

assumptions (3)

domain assumption Asymptotic power series are closed under addition, multiplication, and reciprocation, and the asymptotic expansion of a reciprocal is the formal reciprocal of the asymptotic expansion.
Used in the proof of Theorem 2.7 when substituting A(1/x) ~ x e^{-x} Ei(x) into an exact algebraic identity and taking reciprocals; standard in asymptotic analysis but not stated or proved in the paper.
standard math Standard asymptotic expansion of the exponential integral: Ei(x) ~ e^x/x sum_{n>=0} n!/x^n as x -> infinity.
Cited as Proposition 2.4 from NIST DLMF and used in the proof of Theorem 2.7.
standard math Classical estimate of de la Vallee Poussin: pi(x) = li(x) + O(x exp(-a sqrt(log x))) for some a > 0.
Used in the remark after Proposition 1.2 to justify that 1/pi(x) and 1/li(x) have the same asymptotic expansion; not central to the main theorem.

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Cite this review

Pith. "Pith review of Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals." pith.science (2026). https://pith.science/paper/2HTN6L2G

@misc{pith2026241219866,
  author       = {Pith},
  title        = {Pith review of: Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HTN6L2G}},
  note         = {Machine review of arXiv:2412.19866}
}

read the original abstract

Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function, obtaining asymptotic expansions salient to the first Hardy-Littlewood conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

[1]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

4, 430–479

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stepha n Ramon Garcia, The Bateman-Horn conjec- ture: Heuristic, history, and applications , Expositiones Mathematicae 38 (2020), no. 4, 430–479

work page 2020

[2] [2]

Comtet, Sur les coeﬃcients de l’inverse de la s´ erie formelle ∑ n!xn, C

L. Comtet, Sur les coeﬃcients de l’inverse de la s´ erie formelle ∑ n!xn, C. R. Acad. Sci. Paris S´ er. A-B 275 (1972), A569–A572

work page 1972

[3] [3]

Charles Jean de la Vall´ ee Poussin, Sur la fonction zeta de riemann et le nombre des nombres premi ers inferieur a une limite donn´ ee, M´ emoires couronn´ es de L’Acad´ emie de Belgique59 (1899)

work page

[4] [4]

NIST Digital Library of Mathematical Functions . F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

[5] [5]

Jesse Elliott, Asymptotic expansions of the prime counting function , Journal of Number Theory 238 (2022), 353–395

work page 2022

[6] [6]

Gao, Sergey Kitaev, and Philip B

Alice L.L. Gao, Sergey Kitaev, and Philip B. Zhang, On pattern avoiding indecomposable permutations , Integers 18 (2018)

work page 2018

[7] [7]

G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expressi on of a number as a sum of primes , Acta Mathematica 44 (1923), 1–70

work page 1923

[8] [8]

Hunter, Asymptotic Analysis and Singular Perturbation Theory , 2004

John K. Hunter, Asymptotic Analysis and Singular Perturbation Theory , 2004. University of California at Davis

work page 2004