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On the irrationality of certain $p$-adic zeta values

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Pith's one-line read For every prime $p \ge 5$, at least one of the $p$-adic zeta values $\zeta_p(3), \zeta_p(5), \ldots, \zeta_p(c_p)$ is irrational, where $c_p \approx p + (\gamma+o(1))p/\log p$.

desk verdict Solid new p-adic analogue of Zudilin for every prime p ≥ 5; the main theorem holds, and the two 'elementary' checks should be written out before publication. read the letter →

arxiv 2505.23088 v1 pith:VOYRB4UN submitted 2025-05-29 math.NT

Li Lai , Cezar Lupu , Johannes Sprang This is my paper · ORCID
classification math.NT MSC 11J7211M0633C20
keywords p-adiczetavalueirrationalityKubota–LeopoldtL-functionhypergeometricseriesVolkenbornintegralcriterionlinearformsprimenumbertheorem
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 proves a $p$-adic analogue of the classical result that at least one of the odd zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. For every prime $p \ge 5$, it shows that among the $p$-adic zeta values $\zeta_p(3), \zeta_p(5), \ldots, \zeta_p(c_p)$ at least one is irrational, where $c_p$ is an explicit bound depending only on $p$. The bound satisfies $c_p = p + (\gamma+o(1))p/\log p$ as $p \to \infty$, and the weaker Corollary 1.2 gives the uniform estimate $c_p \le p + p/\log p + 5$ for all $p \ge 5$. This matters because, although earlier work showed that infinitely many odd-indexed $p$-adic zeta values are irrational, no unconditional statement of this finite-range form was known for a general prime $p \ge 5$.

What carries the argument

The engine is the rational function $R_n(t) = p^{pn}\, n!^s\, t^{M_0}\ \prod_{j=1}^{p-1}(t+j/p)^n / (t)_{p-1+s}^{n+1}$, where $(t)_m$ is the rising factorial (Pochhammer symbol). Its partial-fraction coefficients $r_{i,k}$ are assembled, through Volkenborn integrals (the $p$-adic integral over $\mathbb{Z}_p$) applied to a primitive of $R_n$, into the linear forms $S_n$ in $1$ and the $p$-adic zeta values. The arithmetic step shows that after multiplying by $d_n^{p-1+s}$ and dividing by the common divisor $\Phi_n = \prod_{\sqrt[p]{n}<q\le n} q^{\varphi(n/q)}$, where $\varphi$ is a step function of the fractional part of $n/q$, every coefficient becomes an integer; the key lower bound on $q$-adic valuations rests on a floor-function inequality (Lemma 5.4). The Archimedean growth of $\Phi_n$ is then computed with the prime number theorem, and the $p$-adic size of $S_n$ is computed exactly on a chosen subsequence, so the two estimates balance to satisfy the irrationality criterion.

What would settle it

Check the inequality of Lemma 5.4 directly for a fixed prime $p \ge 5$ by enumerating the finitely many cells of the fractional-part partition of $(x,y) \in [0,1)^2$; a single violation would break the divisibility of the normalized linear forms. Alternatively, compute the $q$-adic valuation of a specific coefficient $r_{p-1+s,k}$ for a prime $q$ with $\sqrt[p]{n}<q\le n$ and compare it with $\varphi(n/q)$; a mismatch would disprove the lemma.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any prime $p \ge 5$ there exists an odd integer $i$ in the interval $[3, c_p]$ such that the $p$-adic zeta value $\zeta_p(i)$ is irrational, with $c_p$ given explicitly in terms of the digamma function and Euler's constant. A weaker consequence, Corollary 1.2, replaces $c_p$ by the uniform bound $p + p/\log p + 5$. The proof constructs, for each $n$ in an infinite set $I$, a nonzero linear form $S_n = \rho_0 + \sum_{3 \le i \le p-1+s,\ i\ \mathrm{odd}} \rho_i\, p^i \zeta_p(i)$, whose coefficients come from the partial fractions of an explicit rational function. After rescaling by $\Phi_n^{-1} d_n^{p-1+s}$, all coefficients $\rho_i$ become integers; simultaneously $|S_n|_p \to 0$ while the rescaled coefficients remain controlled in the Archimedean absolute value. An elementary $p$-adic irrationality criterion (Lemma 2.1) then forces at least one of the $\zeta_p(i)$ in the sum to be irrational.

Load-bearing premise

The load-bearing premise is the floor-function inequality in Lemma 5.4, stated as elementary to check but not proved: for all real $x,y$, $\lfloor py\rfloor + \lfloor px-py\rfloor - p\lfloor y\rfloor - p\lfloor x-y\rfloor \ge \varphi(x)$, where $\varphi$ is the step function in (5.4); if this inequality fails for some $p \ge 5$, the common divisor $\Phi_n$ would not divide all coefficients and the irrationality criterion could not be applied.

Editorial extensions

If this is right

  • For each prime $p \ge 5$, at least one of the finitely many values $\zeta_p(3), \zeta_p(5), \ldots, \zeta_p(c_p)$ is irrational.
  • The guaranteed irrational value lies among odd indices no larger than $p + p/\log p + 5$.
  • As $p$ grows, the length of the interval containing the irrational value is $(\gamma+o(1))p/\log p$, which is much smaller than $p$ itself.
  • The theorem does not settle the irrationality of $\zeta_p(3)$ for a general prime $p \ge 5$; it only shows that the obstruction is located at or below the explicit bound $c_p$.

Reading between the lines

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

  • Varying the integer parameter $s$ or the shift in the rational function would likely change the constant in front of $p/\log p$; a natural testable extension is to look for the shortest interval that can be forced for each $p$ by optimizing these choices.
  • Because the constant in $c_p$ comes from the prime number theorem through $\Phi_n$, any future improvement in prime-distribution estimates would automatically sharpen the interval, while the method itself does not depend on the Riemann hypothesis.
  • The explicit linear forms could in principle be evaluated numerically for small primes (for instance $p=5$) to high $p$-adic precision, giving concrete evidence about which value in the interval is irrational.
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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. This paper proves a p-adic analogue of Zudilin's theorem: for every prime p ≥ 5 there is an odd integer i in [3, c_p] such that the p-adic zeta value ζ_p(i) is irrational, where c_p is given explicitly in terms of p and the digamma function (Theorem 1.1). A corollary gives the uniform interval [3, p + p/log p + 5]. The proof constructs rational functions R_n(t), defines linear forms S_n via Volkenborn integrals, proves integrality of normalized coefficients using a common divisor Φ_n, bounds |S_n|_p exactly on a subsequence, and controls Archimedean growth; an elementary irrationality criterion then forces one of the values to be irrational.

Significance. If correct, this is a genuine p-adic counterpart of Zudilin's 2001 result and strengthens the known effective finiteness results for p-adic zeta values. The construction is explicit and parameter-free: all constants are determined by p and standard special functions. The central mechanism, a large common divisor in the coefficients, is the right p-adic analogue of Zudilin's method, and the paper carefully situates itself in the existing literature. The main weaknesses are missing or incorrect proofs of technical supporting statements; none of them appear to affect the truth of the theorem, and all are repairable without changing the architecture of the proof.

major comments (3)
  1. [§5, Lemma 5.4] The floor-function inequality after (5.4), namely ⌊py⌋ + ⌊px−py⌋ − p⌊y⌋ − p⌊x−y⌋ ≥ φ(x), is asserted as 'elementary to check' and is not proved; the references to figures in [23] and [13] are not a substitute for a proof. This inequality is the source of the common divisor Φ_n and therefore underpins the integrality of the normalized coefficients in Lemmas 5.5 and 5.7. Please include a complete proof, for example by writing x = m + a, y = n + b and using ⌊u⌋ + ⌊v⌋ ≥ ⌊u+v⌋ − 1.
  2. [§5, Lemma 5.6] The contradiction argument contains an incorrect valuation inference. If v_q(Φ^{-1} d_n^{p−1+s−i} r_{i,k}) ≥ 0 by Lemma 5.4 and the product Φ^{-1} d_n^{p−1+s} r_{i,k}(ν+j/p)^i has negative q-adic valuation, then the derived inequality is v_q(ν+j/p) < −v_q(d_n), not v_q(ν+j/p) > v_q(d_n) as displayed. In fact each summand in the inner sum is individually integral: for q ≠ p, v_q(ν+j/p) ≥ 0, while for q = p, v_p(ν+j/p) = −1 and v_p(d_n) ≥ 1 for n > (p+s)^4, so v_p of the summand is at least i(v_p(d_n) − 1) ≥ 0. Lemma 5.6 therefore follows directly from Lemma 5.4; please replace the contradiction argument with this direct proof.
  3. [§8, Remark 8.1] The bound 'the greatest odd integer not exceeding c_p ≤ p + p/log p + 5' and the asymptotic c_p = p + (γ+o(1))p/log p are stated without proof. These statements are used in the abstract and in Corollary 1.2, and they are not immediate from Lemma 7.4 as written. Please supply a short derivation, or promote the remark to a lemma with proof.
minor comments (3)
  1. [§6, Lemma 6.2] The sentence identifying k = (p−1)p^N + 1 as the only relevant integer with v_p(h_k/(1−k)) = −N is terse; please spell out the comparison for integers k that are congruent to 1 modulo p−1 but not modulo p, since the displayed congruences (6.13) do not directly control those cases.
  2. [Title/Abstract] The title in the header reads 'certainp-adic' without a space; please fix this and other typographical artifacts in the front matter.
  3. [§5, Lemma 5.3] After equation (5.2), the sum over λ is described as over 'all families of non-negative integers'; it may help readers to explicitly note that the λ_j/p correspond to the p−1 factors F_j/p(t) and that the total degree condition forces only finitely many choices.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central derivation is self-contained, and the self-citations used are standard lemmas whose assumptions do not encode the target result.

full rationale

The paper's main claim, Theorem 1.1, is derived from an explicit construction: rational functions R_n(t), linear forms S_n, integrality lemmas, p-adic norm estimates, and Archimedean estimates. All constants, including c_p and ϖ_p, are defined explicitly from p and standard special functions; no parameter is fitted to the target irrationality. The cited results with overlapping authorship ([12, Lemma 2.1], [13, Lemma 4.2], [15, Lemmas 2.7 and 2.10]) are general-purpose lemmas with stated assumptions that do not include the existence of an irrational p-adic zeta value or the value of c_p. They provide independent support: Lemma 2.1 is a standard irrationality criterion, Lemma 5.2 is an integrality statement for certain polynomials, and Lemmas 2.4/2.5 are standard p-adic integral identities. The floor-function inequality in Lemma 5.4 is asserted as 'elementary to check' with pointers to [23] and [13], but it is a concrete elementary inequality, not an imported ansatz, and it is verifiable independently of the paper's conclusion. No step reduces by construction to its input: the rational functions are new, the linear form S_n is not defined as the target values, and the irrationality conclusion follows from the criterion applied to genuinely estimated quantities. Minor self-citations are present but none is load-bearing in the sense of making the theorem equivalent to an assumed result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on standard lemmas borrowed from the literature (irrationality criterion, integrality lemmas, prime-number-theorem-based distribution of fractional parts, digamma bounds). None are fitted; all are external theorems. No new entities are introduced.

assumptions (5)
  • standard math Lemma 2.1 (irrationality criterion): if integer linear forms tend to 0 p-adically relative to coefficient size and are nonzero, one argument is irrational.
    Cited from [12, Lemma 2.1]; the proof is not reproduced, but the lemma is a standard Liouville-type criterion used in the field.
  • standard math Floor-function inequality: for all real x, y, ⌊py⌋+⌊px−py⌋−p⌊y⌋−p⌊x−y⌋ ≥ φ(x), where φ is defined by (5.4).
    Used in Lemma 5.4 to prove the common divisor Φ_n; the paper refers to [23, Fig. 9,10] and [13, Fig. 1] for the proof and does not include it in the text.
  • standard math Prime number theorem and the asymptotic distribution of fractional parts {n/q} over primes q in (√(Cn), n] (Lemma 7.2).
    Cited from [23, Lemma 4.4]; controls the growth of Φ_n and hence the interval c_p.
  • standard math Digamma function identities and the bounds ψ(1/p) ∈ (−log2−p, −p) and related inequalities (Lemma 7.4).
    Cited from [9, Cor. 3]; ensures ϖ_p lies in the interval that makes s positive and yields Corollary 1.2.
  • standard math Lemmas 5.1 and 5.2 on integrality of derivatives of Pochhammer products.
    Cited from [24] and [13]; used in the denominator bound Lemma 5.3.

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Cite this review

Pith. "Pith review of On the irrationality of certain $p$-adic zeta values." pith.science (2026). https://pith.science/paper/VOYRB4UN

@misc{pith2026250523088,
  author       = {Pith},
  title        = {Pith review of: On the irrationality of certain $p$-adic zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOYRB4UN}},
  note         = {Machine review of arXiv:2505.23088}
}
abstract

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value $\zeta_p(i)$ is irrational.

Discussion (0). Continue with ORCID to comment.

Reference graph

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