{"id":"c8307251-b384-48ca-a979-331cec551a90","arxiv_id":"2505.23088","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every prime p ≥ 5, some p-adic zeta value ζ_p(i) with odd i ≤ p + p/log p + 5 is irrational.","lead":"This paper proves that for every prime number p at least 5, at least one p-adic zeta value with odd index up to roughly p is irrational. It is the p-adic analogue of Zudilin's 2001 theorem on ordinary zeta values, and provides a finite window in which the first irrational value must appear.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No load-bearing error found: the unproved floor-function inequality in Lemma 5.4 is correct, so the central claim is not threatened.","rationale":"The paper's central claim depends on the integrality of the normalized coefficients via Lemma 5.4; the only genuinely unproved step is the floor-function inequality. My independent derivation confirms it, so the concern does not land. The other flagged issue, Remark 8.1, is a routine estimate from Lemma 7.4. I also reviewed the p-adic norm computation in Lemma 6.2 and found only cosmetic sign/unit omissions that do not alter the valuation. Therefore the conditional verdict can stand; writing out the two-line proof and the Remark 8.1 estimate would make the paper self-contained.","tokens_in":104,"tokens_out":59713,"duration_ms":1187277,"concrete_test":"Add the missing proof of the floor inequality in Lemma 5.4: set a={x}, b={y}; show LHS=⌊pb⌋+⌊p{a−b}⌋ and use ⌊u⌋+⌊v⌋≥⌊u+v⌋−1 (plus the b>a case) to conclude LHS≥max(0,⌊p{x}⌋−1)=φ(x). This settles the reader's concern in the affirmative.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption targets the floor inequality in Lemma 5.4, asserted as 'elementary to check' with references to figures in [23] and [13]. I verified the inequality directly: for real x=m+a, y=n+b with a,b∈[0,1), the left side telescopes to ⌊pb⌋+⌊p{a−b}⌋ if b≤a and to p+⌊pb⌋+⌊p{a−b}⌋ if b>a, where {a−b} is the fractional part of a−b. The standard estimate ⌊u⌋+⌊v⌋≥⌊u+v⌋−1 then gives LHS≥max(0,⌊p{x}⌋−1)=φ(x), exactly (5.4). Hence Lemma 5.4 is sound; the only defect is that the proof is omitted rather than false. The same holds for the numerical bound in Remark 8.1, which follows from Lemma 7.4 in a few lines. I also checked the valuation computation in Lemma 6.2: the congruence steps drop harmless p-adic units (h≡±1 and the factor −1/(p−1)), but the leading term is a nonzero p-adic unit and the exact valuation conclusion is unaffected. No load-bearing correctness gap was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16449,"tokens_out":14080,"duration_ms":143607,"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":[{"comment":"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.","section":"§5, Lemma 5.4"},{"comment":"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.","section":"§5, Lemma 5.6"},{"comment":"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.","section":"§8, Remark 8.1"}],"minor_comments":[{"comment":"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.","section":"§6, Lemma 6.2"},{"comment":"The title in the header reads 'certainp-adic' without a space; please fix this and other typographical artifacts in the front matter.","section":"Title/Abstract"},{"comment":"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.","section":"§5, Lemma 5.3"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the proof strategy is sound, but the manuscript currently contains two load-bearing technical gaps: the unproved floor-function inequality in Lemma 5.4 and the erroneous valuation inference in the proof of Lemma 5.6. Both are locally repairable, and the direct argument for Lemma 5.6 is straightforward. The paper fits a general number theory journal well. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper proves the p-adic analogue of Zudilin for every prime p ≥ 5, and the proof is sound. The one thing that will trip a reader is the unproved floor-function inequality in Lemma 5.4. It looks like a hole; it isn't. I went through it: for x=m+a, y=n+b the left side reduces to ⌊pb⌋+⌊p{a−b}⌋ plus possibly p, and the standard floor estimate gives exactly φ(x) from (5.4). So the common divisor Φ_n has the right size. The same for Remark 8.1: the numerical bound follows from Lemma 7.4 in a few lines. The authors should be asked to include those arguments; right now they assert 'elementary to check' and point to figures in [23] and [13], which is not self-contained.\n\nThe construction itself is genuinely new. They build a family of rational functions with denominator (t)_{n+1}^{p-1+s}, extract linear forms in 1 and p-adic zeta values via Volkenborn integrals, and use a p-dependent common divisor Φ_n to force integrality. The payoff is an explicit interval [3, c_p] for every prime p≥5, with c_p = p + (γ+o(1)) p/log p. That sharpens the earlier asymptotic Ball–Rivoal type results and extends the p=2 work of Lai to all primes. No fitted constants anywhere; all parameters are explicit from p and standard special functions.\n\nThe p-adic valuation lemma (Lemma 6.2) is the other dense spot. I checked the congruence step: dropping the p-adic units is legitimate, and the leading term is a unit, so the exact valuation statement is fine. The Archimedean estimates in §7 are routine. The paper leans on prior results (the irrationality criterion from [12], Lemma 7.2 from [23])—normal for this area, and the citations are fair.\n\nBottom line: for a specialist in Diophantine approximation, this is worth careful reading and worth citing. It deserves a serious referee, not a desk reject. My only referee request would be: write out Lemma 5.4 and Remark 8.1, and maybe add a sentence in Lemma 6.2 justifying why the terms dropped do not change the valuation. I would send it out.","headline":"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.","tokens_in":17040,"tokens_out":3057,"would_cite":true,"duration_ms":33642,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J72","11M06","33C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $c_p \\approx p + (\\gamma+o(1))p/\\log p$.","keywords":["p-adic zeta value","irrationality","Kubota–Leopoldt p-adic L-function","hypergeometric series","Volkenborn integral","irrationality criterion","linear forms","prime number theorem"],"falsifier":"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.","tokens_in":16012,"feed_emoji":"🔢","tokens_out":11590,"duration_ms":105986,"temperature":0.7,"pith_summary":"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 $p \\ge 5$.","feed_headline":"Every prime p ≥ 5 has an irrational odd p-adic zeta value","feed_subtitle":"The irrational value sits among odd indices up to about p + p/log p, a finite explicit range.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $c_p$."],"supporting_citations":[{"why":"Supplies the elementary irrationality criterion (Lemma 2.1) that turns vanishing p-adic size of integral linear forms into irrationality of one of the entries.","marker":"[12]"},{"why":"Pioneers the Volkenborn-integral construction of linear forms in p-adic zeta values, which the present proof adapts.","marker":"[21]"},{"why":"Provides the Bernoulli-functional identity (Lemma 2.4) and the p-adic Hurwitz zeta lemma (Lemma 2.5) used to evaluate the Volkenborn integrals as zeta values.","marker":"[15]"},{"why":"Supplies Lemma 5.1, the integrality of normalized derivatives of the Pochhammer reciprocal, used to bound the denominators of the coefficients.","marker":"[24]"},{"why":"Supplies Lemma 5.2, a companion integrality statement, and the floor-function estimation method cited in Lemma 5.4.","marker":"[13]"},{"why":"Provides the figures and method for estimating sums of floor functions that underlie the unproved inequality in Lemma 5.4.","marker":"[23]"},{"why":"The classical theorem that one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational, whose p-adic analogue is established in this paper.","marker":"[22]"}],"fun_headline_variants":["Odd p-adic zeta values: always an irrational in range","Every prime ≥5 has an irrational p-adic zeta value","p-adic zeta: irrational odd index exists for each prime","Irrational p-adic zeta within small odd index range"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Odd p-adic zeta values: always an irrational in range","Every prime ≥5 has an irrational p-adic zeta value","p-adic zeta: irrational odd index exists for each prime","Irrational p-adic zeta within small odd index range"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000124,"raw_usage":{"total_tokens":1085,"prompt_tokens":910,"completion_tokens":175,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":100}},"tokens_in":526,"tokens_out":175,"duration_ms":2505,"temperature":1.0,"reasoning_tokens":100,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:56:03.579720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Lai,On the irrationality of certain2-adic zeta values, Int","cited_arxiv_id":null,"evidence_quote":"Supplies the elementary irrationality criterion (Lemma 2.1) that turns vanishing p-adic size of integral linear forms into irrationality of one of the entries."},{"cited_title":"Sprang,Linear independence result forp-adicL-values, Duke Math","cited_arxiv_id":null,"evidence_quote":"Pioneers the Volkenborn-integral construction of linear forms in p-adic zeta values, which the present proof adapts."},{"cited_title":"Many $p$-adic odd zeta values are irrational","cited_arxiv_id":"2306.10393","evidence_quote":"Provides the Bernoulli-functional identity (Lemma 2.4) and the p-adic Hurwitz zeta lemma (Lemma 2.5) used to evaluate the Volkenborn integrals as zeta values."},{"cited_title":"Zudilin,Arithmetic of linear forms involving odd zeta values, J","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 5.1, the integrality of normalized derivatives of the Pochhammer reciprocal, used to bound the denominators of the coefficients."},{"cited_title":"Small improvements on the Ball-Rivoal theorem and its $p$-adic variant","cited_arxiv_id":"2407.14236","evidence_quote":"Supplies Lemma 5.2, a companion integrality statement, and the floor-function estimation method cited in Lemma 5.4."},{"cited_title":"Zudilin,Irrationality of values of the Riemann zeta function, Izvestiya Ross","cited_arxiv_id":null,"evidence_quote":"Provides the figures and method for estimating sums of floor functions that underlie the unproved inequality in Lemma 5.4."},{"cited_title":"Zudilin,One of the numbersζ(5),ζ(7),ζ(9),ζ(11)is irrational, Uspekhi Mat","cited_arxiv_id":null,"evidence_quote":"The classical theorem that one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational, whose p-adic analogue is established in this paper."}],"review_version":1}