REVIEW 4 major objections 5 minor 22 references
On non-holonomicity, transcendence and $p$-adic valuations
T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The power series of q-adic valuations, and their reductions modulo k, are non-holonomic for every q,k ≥ 2, and take transcendental values at infinitely many rational and algebraic inputs.
desk verdict Non-holonomicity is new and likely correct; the transcendence proof has a real but repairable gap in the "positive segments" hypothesis. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity writes $V_q(X)$ as a sum of geometric blocks, $V_q(X)=\sum_{m\ge 1} X^{q^m}/(1-X^{q^m})$, grouping terms by valuation level so that each coefficient becomes a count of blocks; the reduced series $V_{q,k}$ is the same sum with block coefficients $a_j=1$ for $k\nmid j$ and $a_j=1-k$ for $k\mid j$. Non-holonomicity is obtained in two cases: if the series were rational, the valuation sequence would satisfy a constant-coefficient recurrence, which is contradicted by a pair of integers with valuations k and k+1 whose preceding d neighbors have identical valuations; if it is not rational, a classical dichotomy for integer-coefficient series says the unit circle is a natural boundary, and a holonomic series admits analytic continuation around finitely many singularities, so it cannot have such a boundary. For transcendence, the block structure gives explicit rational approximants $A_n/B_n$ with $B_n=b^{q^n}-a^{q^n}$, and Roth's rational-approximation theorem for algebraic numbers applies once the error term is dominated by $1/B_n^{2+\delta}$; the hypothesis $q>2\log b/(\log b-\log a)$ is precisely the condition that makes this domination possible. The roots-of-unity extension uses the identity that $V_q(\omega X)-V_q(X)$ is rational for $\omega$ of order $q^\ell$, so an algebraic value at $\omega a/b$ would force one at $a/b$.
What would settle it
Run a recurrence-guessing routine on the first million terms of $\nu_2(n)$: the paper predicts that no linear recurrence with polynomial coefficients of any fixed order and degree is satisfied, so a recurrence that validates on a long tail of terms would refute Theorem 1. Separately, evaluate $V_3(1/2)$ to several hundred digits and run LLL-based algebraic-number recognition: the paper predicts a transcendental number, so any consistent algebraic candidate of degree below 30 with small height would refute Theorem 2.
Extended reading notes
Core claim
For integers q,k ≥ 2 let ν_q(n) be the exponent of the largest power of q dividing n. The paper proves that the formal power series $V_q(X)=\sum_{n\ge 1}\nu_q(n)X^n$ and $V_{q,k}(X)=\sum_{n\ge 1}(\nu_q(n)\bmod k)X^n$ are non-holonomic over $\mathbb{C}$, meaning neither satisfies any linear differential equation with polynomial coefficients, and that neither is algebraic over the rational function field of any characteristic-0 field whose elements are algebraic numbers. It then proves that whenever a and b are coprime with a < b and q > 2 log b/(log b − log a), the real numbers $V_q(a/b)$ and $V_{q,k}(a/b)$ are transcendental, and the same holds for the complex values $V_q(\omega a/b)$ and $V_{q,k}(\omega a/b)$ where $\omega$ is any root of unity of order $q^\ell$. These theorems are corollaries of a more general statement about series $W(X)=\sum_j a_j X^{q^j}/(1-X^{q^j})$ with bounded integer coefficients that are not rational and have positive segments. Finally, the mod-k valuation sequence is shown to be w-automatic, so over finite fields its generating series is algebraic, while the characteristic-0 series remains non-holonomic.
Load-bearing premise
The argument that the series cannot be rational depends on the unproved existence, for every recurrence order d, of two integers with q-adic valuations k and k+1 whose preceding d neighbors have identical valuations, and if that combinatorial claim failed the non-holonomicity and the transcendence at roots of unity would lose their footing.
Editorial extensions
If this is right
- No finite-order differential equation with polynomial or rational coefficients has $V_q$ or $V_{q,k}$ as a solution, so the valuation series cannot be captured by any classical D-finite framework.
- There are infinitely many rational inputs $a/b$ and infinitely many algebraic-irrational inputs $\omega a/b$ at which both series take transcendental values.
- Over every characteristic-0 field A of algebraic numbers, neither series is algebraic over $A(X)$, ruling out polynomial equations of any degree with coefficients in $A(X)$.
- The mod-$k$ valuation sequence is $w$-automatic; in positive characteristic its generating series is algebraic, while in characteristic zero it is not even algebraic, so automaticity does not tame the generating series across characteristics.
- The period-doubling sequence serves as the explicit example: its generating series is non-holonomic over $\mathbb{C}$.
Reading between the lines
- Beyond the paper: the threshold in (3) comes from the exponent 2 in Roth's theorem, so any future improvement in uniform rational-approximation bounds would widen the transcendence range, possibly to all $a/b<1$ with the same $q$.
- Beyond the paper: the same block decomposition with bounded integer coefficients is a template, so alternating or truncated variants such as $\sum_j (-1)^j X^{q^j}/(1-X^{q^j})$ are natural test cases that should inherit the same non-holonomicity and transcendence.
- Beyond the paper: the contrast between algebraicity over finite fields and non-algebraicity over characteristic-0 fields suggests the obstruction is characteristic-specific; one could test whether other automatic sequences, for instance sum-of-digits sequences, display the same split.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the generating series V_q(X)=sum_{n>=1} nu_q(n)X^n and V_{q,k}(X)=sum_{n>=1}(nu_q(n) mod k)X^n for integer q>=2. Theorem 1 claims that both series are non-holonomic over C, Theorem 2 claims that their values at rationals a/b<1 satisfying q>2 log b/(log b-log a), and at roots-of-unity twists of such rationals, are transcendental, and Theorem 3 claims non-algebraicity over any characteristic-zero field of algebraic numbers. The proofs use the Polya-Carlson theorem for the non-holonomicity, an explicit block decomposition of the series, and Roth's theorem for the transcendence statements. The final sections apply the results to q-automatic sequences, including the period-doubling sequence.
Significance. If the main results are correct, they are natural and interesting: the series of q-adic valuations are shown to lie far outside the holonomic world, and their values are transcendental at an explicit infinite family of rational inputs. The block decompositions (4) and (8) are explicit and checkable, and the applications to automatic sequences give a clean connection to known phenomena. The paper contains no fitted parameters or circular derivations; the external theorems used are standard. However, several load-bearing points in the proofs are not established as written, so the significance can only be assessed after those gaps are repaired.
major comments (4)
- [Section 3, definition of positive segments; Proposition 3.3] The definition of 'positive segments' requires B_j(x)<B_{j+1}(x) for all x in (0,1), but for V_q the proposed blocks B_j=f_j(x) satisfy f_j(x)>f_{j+1}(x), and for V_{q,k} the paper itself proves A_{k,j}(x)>A_{k,j+1}(x) immediately before Proposition 3.3. Thus neither series satisfies the hypothesis of Theorem 5 as stated, so the invocation of Theorem 5 in Section 4.3 is blocked. Since the proof of Theorem 5 only uses positivity of the blocks, the likely fix is to weaken the hypothesis to B_j(x)>0 and to prove explicitly that the partial sums at segment endpoints are strictly increasing; this must be stated and proved.
- [Section 4.2, Theorem 5 and Section 4.1] The rational approximants A_n/B_n defined in (14) are not shown to be in lowest terms. Theorem 4 is only a valid form of Roth's theorem for reduced fractions; as stated it is false for arbitrary non-reduced representations, because scaling a convergent p_n/q_n by a large common factor c_n gives A_n/B_n=c_n p_n/c_n q_n with |alpha-A_n/B_n|<1/B_n^{2+delta} for any irrational alpha once c_n is large enough. Since B_n may share a large factor with A_n, inequality (15) does not imply the existence of reduced denominators satisfying the Roth bound. The proof must either show that the reduced denominator of A_n/B_n grows like B_n up to a fixed constant, or replace the argument by one adapted to the specific structure of these approximants. This gap is load-bearing for Theorem 2.
- [Section 4.4, proof of Theorem 6] The specialization argument for q=2 is impossible: the proof needs a rational a/b in (0,1) satisfying q>2 log b/(log b-log a), but for q=2 no positive integers a<b satisfy this strict inequality, since 2 log b/(log b-log a) is always at least 2 (with equality only approached as a/b tends to 0). Therefore Theorem 6 is not proved for q=2, and since Theorem 3 covers q=2, this case needs a separate argument, for example deriving non-algebraicity from the non-holonomicity in Theorem 1, or a different specialization argument.
- [Proposition 2.1, Case 1, and Proposition 2.2] The proof asserts the existence of m1,m2 with nu_q(m1)=k, nu_q(m2)=k+1 (or with different residues modulo k in Proposition 2.2) and nu_q(m1-i)=nu_q(m2-i) for i=1,...,d, but no construction is given. This existence is load-bearing for the rational case. The assertion is true: taking m1=q^k u and m2=q^{k+1}v with gcd(uv,q)=1, q^k>d, and m1,m2>N+d gives nu_q(m1-i)=nu_q(i)=nu_q(m2-i) for 1<=i<=d. The manuscript should include this construction explicitly and also ensure m1-d and m2-d are at least N.
minor comments (5)
- [Section 4.3] The heading 'Proof of Theorem 3' actually proves Theorem 2; the generalized non-algebraicity statement is proved later in Section 4.4 as Theorem 6.
- [Equation (11)] The error term is written E_{p,n}(a,b) in (11) but E_{q,n}(a,b) in the following lines; the notation should be unified.
- [Section 4.2] The sentence 'the corresponding queue is positive' should read 'tail', and the later remark that transcendence holds 'except for at most a finite number of integers q' is confusing because the condition (3) already specifies which q are covered.
- [Theorem 4] The statement of Roth's theorem should explicitly require gcd(A_n,B_n)=1 and use the reduced denominator; otherwise the theorem as printed is not a true statement.
- [Section 5, Theorem 7] In the definition of the automaton, 'the state set S (also called the input alphabet)' should be 'the input alphabet is {0,...,w-1}'; the current phrasing conflates the state set and the alphabet.
Circularity Check
No significant circularity: the main theorems are derived from external results (Pólya–Carlson, Roth, Christol, Denef–Lipshitz) and the self-citations are motivational, not load-bearing.
full rationale
The derivation chain is self-contained in the sense required by the circularity check. Theorem 1 is proved from the Pólya–Carlson theorem and elementary spacing properties of q-adic valuations, both external to the paper's own claims. Theorem 2 is proved via Roth's theorem applied to rational approximants obtained from the series' block decomposition, and Theorem 3/6 is reduced to Theorem 5 by a specialization argument. The self-citations ([15], [16], [17]) concern arithmetic-term representations and appear only as introductory motivation or illustrative context, not as hypotheses in the proofs of non-holonomicity or transcendence. No parameter is fitted to data, and no quantity called a prediction is defined in terms of a target result. The paper does contain apparent mathematical gaps that are visible from its own text, such as Proposition 3.3 claiming 'positive segments' for V_q and V_{q,k} even though the paper observes f_n(x) > f_{n+t}(x), which implies B_j > B_{j+1}; and Theorem 5's application of Roth does not explicitly establish distinctness of the approximants A_n/B_n. These are correctness risks, not circular reductions: the intended conclusions are not assumed as inputs, and no equation or definition forces the conclusion to equal its hypothesis. Under the hard rules requiring an exhibited reduction, no circular step is present.
Assumptions & free parameters
assumptions (5)
- standard math Polya-Carlson theorem: an integer coefficient power series with radius 1 is either rational or has the unit circle as a natural boundary.
- standard math Roth's theorem on rational approximations to algebraic numbers.
- standard math Holonomic power series can be analytically continued except at finitely many singularities (roots of the leading coefficient polynomial).
- ad hoc to paper Existence of m1,m2 with nu_q(m1)=k, nu_q(m2)=k+1 and matching q-adic valuation windows for the preceding d integers.
- standard math Christol's theorem and Denef-Lipshitz theorem on automatic sequences and algebraic power series over finite fields and p-adic integers.
Cite this review
Pith. "Pith review of On non-holonomicity, transcendence and $p$-adic valuations." pith.science (2026). https://pith.science/paper/LIHXEYOJ
@misc{pith2026241216517,
author = {Pith},
title = {Pith review of: On non-holonomicity, transcendence and $p$-adic valuations},
year = {2026},
howpublished = {\url{https://pith.science/paper/LIHXEYOJ}},
note = {Machine review of arXiv:2412.16517}
}
abstract
Let ${\nu}_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients ${\nu}_q(n)$, respectively ${\nu}_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence.
Reference graph
Works this paper leans on
-
[1]
Jean-Paul Allouche and Jeffrey Shallit. Automatic sequences. Theory, applications, generalizati ons. Cambridge: Cambridge University Press, 2003. doi:10.1017/CBO9780511546563. (page 11)
-
[2]
Ludwig Bieberbach. Funktionentheorie. Johnson, New York, 1968. (page 3)
work page 1968
-
[3]
¨Uber Potenzreihen mit ganzzahligen Koeffizienten
Fritz Carlson. ¨Uber Potenzreihen mit ganzzahligen Koeffizienten. Math. Z. , 9:1–13, 1921. doi:10.1007/BF01378331. (page 3)
-
[4]
G. Christol, T. Kamae, Michel Mend` es France, and G´ erar d Rauzy. Suites alg´ ebriques, automates et substitutions. Bull. Soc. Math. Fr. , 108:401–419, 1980. doi:10.24033/bsmf.1926. (page 11)
-
[5]
Algebraic power series a nd diagonals
Jan Denef and Leonard Lipshitz. Algebraic power series a nd diagonals. J. Number Theory , 26:46–67,
-
[6]
On t he non-holonomic character of logarithms, pow- ers, and the nth prime function
Philippe Flajolet, Stefan Gerhold, and Bruno Salvy. On t he non-holonomic character of logarithms, pow- ers, and the nth prime function. The electronic journal of combinatorics , 11(2), (The Stanley Festschrift volume):1–16, 2005. doi:10.37236/1894. (page 3)
-
[7]
On some non-holonomic sequences
Stefan Gerhold. On some non-holonomic sequences. Electron. J. Combin. , 11(1):Research Paper 87, 8,
-
[8]
Applied and computational complex analysis, vol
Peter Henrici. Applied and computational complex analysis, vol. 2 . John Wiley, New York, 1974. (page 4)
work page 1974
Show all 22 references
-
[9]
The concrete tetrahedron
Manuel Kauers and Peter Paule. The concrete tetrahedron. Symbolic sums, recurrence equat ions, gen- erating functions, asymptotic estimates . Texts Monogr. Symb. Comput. New York, NY: Springer, 2011. doi:10.1007/978-3-7091-0445-3 . (page 3)
2011 doi
-
[10]
S. S. Marchenkov. Superpositions of elementary arithm etic functions. Diskretn. Anal. Issled. Oper., Ser. 1, 13(4):33–48, 2006. (page 1)
2006
-
[11]
Plain bases for classes of primitive recursive functions
Stefano Mazzanti. Plain bases for classes of primitive recursive functions. MLQ Math. Log. Q. , 48(1):93– 104, 2002. doi:10.1002/1521-3870(200201)48:1<93::AID-MALQ93>3. 0.CO;2-8 . (page 1)
2002 doi
-
[12]
Published electronically at http://oeis.org
OEIS Foundation IncThe On-Line Encyclopedia of Intege r Sequences, 2023. Published electronically at http://oeis.org. (page 12)
2023
-
[13]
¨Uber Potenzreihen mit ganzzahligen Koeffizienten
Georg P´ olya. ¨Uber Potenzreihen mit ganzzahligen Koeffizienten. Math. Ann. , 77(4):497–513, 1916. doi:10.1007/BF01456965. (page 3)
1916 doi
-
[14]
On other two representations of the C-r ecursive integer sequences by terms in modular arithmetic, 2024
Mihai Prunescu. On other two representations of the C-r ecursive integer sequences by terms in modular arithmetic, 2024. arXiv:2406.06436. (page 3)
2024 arXiv
-
[15]
On the re presentation of C-recursive integer sequences by arithmetic terms, 2024
Mihai Prunescu and Lorenzo Sauras-Altuzarra. On the re presentation of C-recursive integer sequences by arithmetic terms, 2024. arXiv:2405.04083. (pages 2, 3)
2024 arXiv
-
[16]
On the re presentation of number-theoretic functions by arithmetic terms, 2024
Mihai Prunescu and Lorenzo Sauras-Altuzarra. On the re presentation of number-theoretic functions by arithmetic terms, 2024. arXiv:2407.12928. (page 1)
2024 arXiv
-
[17]
Arithmetic-term rep resentations for the greatest common divisor,
Mihai Prunescu and Joseph Shunia. Arithmetic-term rep resentations for the greatest common divisor,
-
[18]
Klaus F. Roth. Rational approximations to algebraic nu mbers. Mathematika, 2:1–20, 1955. doi:10.1112/S0025579300000644. (page 7)
1955 doi
-
[19]
Simion Stoilow
Richard P. Stanley. Enumerative combinatorics. Vol. 1. , volume 49 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2nd ed. edition, 20 12. URL: www.cambridge.org/de/knowledge/isbn/item6832283/?site_locale=de_DE. (page 3). ON NON-HOLONOMICITY, TRANSCENDENCE AND...
- [1987]
- [2004]
- [2024]
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