RESULTS: Robin's inequality beyond the verified range (`robin_tfree/`)

Status: candidate, ordinary derivation, unreviewed, pending external verification. The numerical step is enclosure-carrying (Arb balls, cross-checked by a second quadrature route); the inputs are published theorems and tables, quoted in section 3 and not recomputed here. The steps that most need outside eyes are Lemma 3(b),(c) (the reduction) and the use of Buthe's Table 1 and BKLNW's Tables 8 and 9 (section 9). No RH claim. Against the literature found by the search of section 9, the t-free statement is a delta of four (25 against the published 21); that is a recorded search, not a novelty claim.

Summary

Robin (1984): RH holds iff sigma(n) < e^gamma n log log n for every n > 5040. Morrill and Platt verified the inequality for every 5040 < n <= x0#, x0 = 29 996 208 012 611. Every known t-free and valuation result above that range reduces to one quantity at the primorials,

E(x) = log( e^{-gamma} prod_{p <= x} p/(p-1) / log theta(x) ), x = p_k >= x0.

Claim 1 (candidate, ordinary derivation, unreviewed, pending external verification). For every x >= x0, E(x) <= 2.4807 * 10^-8 (route A, which uses only Buthe's x - theta(x) <= 1.95 sqrt(x)), and E(x) <= 2.3414 * 10^-8 (route B, Buthe's dyadic Table 1).

Corollaries (candidate, same grade).

1. Robin's inequality holds for every 25-free n > 5040. Best published t-free result found (section 9): 21-free (Axler, Ramanujan J. 61 (2023) 909-919; reproved by Assani, Chester, Paschal, Integers 26 (2026) A86).
2. It holds for every n > 5040 with q^(nu_q(n)+1) < Q* for some prime q, where Q* >= 40 311 388 (route A; 42 709 615 route B). Spelled out: nu_2(n) <= 24, or nu_3 <= 14, nu_5 <= 9, nu_7 <= 7 (8 by route B), nu_11 <= 6, nu_q <= 5 for q in {13, 17}, <= 4 for 19 <= q <= 31, <= 3 for 37 <= q <= 79, <= 2 for 83 <= q <= 337 (349 by route B), <= 1 for primes q < 6349, or q does not divide n for some prime q < 4.03 * 10^7. Axler's Theorem 3 (Theorem 1.4 in arXiv:2110.13478v3) is the same shape; his list (nu_2 <= 20, nu_5 <= 8, ..., nu_q = 1 up to 1777) corresponds to a threshold between 1777^2 and 1783^2, about 3.17 * 10^6 (pinned).
3. sigma(n) < (1 + 2.49 * 10^-8) e^gamma n log log n for every n > 5040 (Axler 2023, Corollary 2, which is Corollary 3.1 in arXiv v3: 1 + 3.15367 * 10^-7; route B gives 2.35 * 10^-8). Axler and Nicolas, Acta Arith. 209 (2023) 357-383, give further effective upper bounds for n/phi(n) and sigma(n)/n that this hunt has not read (paywalled), so this comparison is against Axler 2023 only.
4. Hence RH holds iff Robin's inequality holds for every n > 5040 divisible by M* = prod_{q < Q*} q^(e_q), e_q the least nonnegative integer e with q^(e+1) >= Q*. This does not reduce the remaining problem to 25-full integers: a multiple of M* may have prime factors with exponent less than 25.

What makes the difference is one exact identity (section 2): the boundary term of Mertens' partial summation, R(x)/(x log x), and the denominator correction log(log theta(x)/log x) cancel to first order. Morrill-Platt (v4, Lemma 10) and Axler (proof of Theorem 1) bound them separately. At x0 that costs 2 * 1.95/(sqrt(x0) log x0) = 2.29 * 10^-8, and with that cost added route A gives 24, not 25 (pinned by test_separate_bounding_loses_what_the_cancellation_keeps).

1. Setting

theta(x) = sum_{p <= x} log p, psi the Chebyshev function, N_k = p_1 ... p_k, so log N_k = theta(p_k) and N_k/phi(N_k) = prod_{p <= p_k} p/(p-1). Then

N_k/phi(N_k) = e^gamma log log N_k exp(E(p_k)). (1.1)

Nicolas (1983) proved that RH holds iff E(p_k) > 0 for every k >= 2. The logarithm defining E(2) is not real, so k = 1 is excluded. This hunt bounds E from above, which is what Robin-type results for restricted families need.

2. The identity

Write R(u) = theta(u) - u, w(u) = (1 + log u)/(u^2 log^2 u), h(p) = -log(1 - 1/p) - 1/p > 0, L = log x, d = R(x)/x, and g(d) = d/L - log(1 + log(1 + d)/L).

Lemma 1 (identity). For real x >= 3,

E(x) = - int_x^inf R(u) w(u) du - sum_{p > x} h(p) + g(d). (2.1)

Proof. Stieltjes integration of 1/(u log u) against d theta gives sum_{p <= x} 1/p = log log x + M + R(x)/(x log x) - int_x^inf R w, with M the Meissel-Mertens constant (the integral converges by the prime number theorem; this is Rosser-Schoenfeld 1962 (4.20) with the remainder kept exact). Mertens' constant relation gamma = M + sum_p h(p) gives sum_{p <= x} log(p/(p-1)) = gamma + log log x + R(x)/(x log x) - int_x^inf R w - sum_{p>x} h(p). Finally log log theta(x) = log log x + log(1 + log(1 + d)/L). Subtracting, R(x)/(x log x) = d/L combines with the last term into g(d). []

Lemma 2 (second order). If |d| <= 10^-6 and L >= 31, then |g(d)| <= d^2/L.

Proof. g(0) = g'(0) = 0 and g''(d) = (L + 1 + l)/((1 + d)^2 (L + l)^2) with l = log(1 + d), so 0 < g'' <= (L + 1 + 1.1e-6)/((1 - 1e-6)^2 (L - 1.1e-6)^2) < 2/L for L >= 31; Taylor's theorem gives |g(d)| <= (sup g''/2) d^2 <= d^2/L. []

Both are checked on real primes in test_bound.py: the partial-summation identity between x < y to 10^-25 at two ranges (and a flipped sign misses by more than 10^-12), the difference form E(y) - E(x) = int_x^y R w + sum_{x<p<=y} h(p) + g(d_y) - g(d_x) to 10^-25, and Lemma 2 at 400 random points.

3. Inputs (quoted, not recomputed)

The printed exponent 13.11485 is rounded down: 10^(10^13.11485) is below x0#, so the top of the verified range is Corollary 2, not Theorem 5 (pinned by test_verified_range_top_is_the_corollary_not_the_rounded_theorem).

Table 1 was transcribed from the published table and compared with a rendering of the page; its maximum equals the 0.94 Buthe states as (1.5), which a test pins. Each Table 1 entry gets +0.005 (the table prints two decimals) and each Table 8/9 entry is inflated by a relative 10^-5 (the paper notes printf rounding of the last digit). Table 8 was computed by its authors assuming RH to height 2.446 * 10^12 (Wedeniwski's H0 = 2 445 999 556 030, their B.1), inside Platt and Trudgian's verification to 3 * 10^12 (Bull. LMS 53 (2021)). Its rows b <= 2000 are computed by the method of Buthe 2016, Theorem 1, which BKLNW restate as their Theorem 16 with a term their footnote 4 says the original statement omitted. That the table values were computed with the corrected term is the authors' statement; this hunt has not checked it, and rows b = 40, 45, 50, all from that method, carry 99% of the tail (pinned).

Transcription check, 2026-10-10. The transcribed values were compared mechanically with text extracted from the freely available arXiv versions: Buthe arXiv:1511.02032v2, Table 1 (all 31 M+ entries) and Theorem 2 (1.5)-(1.7); BKLNW arXiv:2002.11068v2, Table 8 (all 50 rows with b >= 40, the rows used) and Table 9 (A_1(25000)). Every value agrees. The published Math. Comp. tables were not re-read in this check; the Morrill-Platt statements were checked against the Integers PDF and arXiv v4; Rosser-Schoenfeld's Theorem 13 was not rechecked. A transcription that agrees with a preprint is not a review of the tables: their computation, and BKLNW's partial-RH input, await outside review like the rest of the derivation.

4. Reduction from Robin's inequality to E

Lemma 3. Let n > x0#, and let k satisfy N_k <= n < N_(k+1). Then p_k >= x0, omega(n) <= k, log log n >= log theta(p_k), and:

(a) sigma(n)/n = (n/phi(n)) prod_{p^a || n} (1 - p^-(a+1)), and n/phi(n) <= N_k/phi(N_k), so for every prime q | n, sigma(n)/(n log log n) <= e^gamma (1 - q^-(nu_q(n)+1)) exp(E(p_k)).

(b) If a prime q < x0 does not divide n, the omega(n) <= k primes of n avoid q, so n/phi(n) <= (1 - 1/q) N_(k+1)/phi(N_(k+1)) and sigma(n)/(n log log n) < e^gamma (1 - 1/q) exp(E(p_(k+1)) + s) with s = 4/(x0 - 1), since log theta(p_(k+1))/log theta(p_k) <= 1 + 2/theta(p_k) by Bertrand and theta(p_k) >= 0.99 p_k.

(c) If n is t-free (Solé-Planat), sigma(n) <= Psi_t(n) and sigma(n)/(n log log n) <= e^gamma zeta(t)^-1 prod_{p > p_k}(1 - p^-t)^-1 exp(E(p_k)), and log prod_{p > p_k}(1 - p^-t)^-1 <= 2/(x0 - 1) <= s.

These are elementary; the n/phi(n) maximisation and sigma(n)/n < n/phi(n) are checked exhaustively to 2 * 10^5, and Robin's inequality directly on 5041 <= n <= 30030, in test_bound.py; x0 is prime (pinned), so p_k >= x0. Parts (b) and (c) carry the step from E to the integer statements and are the steps a reviewer flagged as most needing outside eyes: ordinary arguments, unreviewed. The checks to 2 * 10^5 exercise only their elementary inequalities at small n; nothing here tests them at n > x0#.

5. The bound

Claim 1 (candidate; restated). For every x >= x0, E(x) <= E* with

Proof. Drop -sum_{p>x} h(p) <= 0 from (2.1). On [x0, 10^19], 0 < u - theta(u) <= B(u) with B(u) = 1.95 sqrt(u) (A) or B(u) = (M+ + 0.005) sqrt(u) + sqrt(u) + u^(1/3) + u^(1/4) + 59 u^(1/5) (B: write u - theta(u) = (u - psi(u)) + sum_{k >= 2} theta(u^(1/k)) and use theta(y) < y; the terms vanish for k > log_2 u <= 63). Past 10^19, |theta(u) - u| <= B(u) = eps(b, b') u + 1.5 sqrt(u), and past e^26000 the Table 9 bound. So -int_x^inf R w <= int_x^inf B w <= int_{x0}^inf B w, the last step because B w >= 0. For Lemma 2, |d| <= 1.957/sqrt(x0) < 10^-6 on [x0, 10^19] (the module's Table 1 envelope; Buthe (1.6) alone gives 1.95) and |d| < 2 * 10^-8 beyond: the largest Table 8 envelope, including 1.5/sqrt(10^19), is less than 1.982 * 10^-8, and the Table 9 envelope is smaller. The test test_far_theta_deficit_envelopes_fit_second_order_domain checks every tail block and the infinite tail. []

Every piece int u^alpha w(u) du has the closed form F(y) = -e^(-beta y)/y - (1 - beta) E1(beta y), beta = 1 - alpha, y = log u (F = log y - 1/y when alpha = 1), evaluated in Arb at 256 bits; the result's radius is below 10^-27. An independent mpmath.quad route over every Table 1 block agrees to a relative 10^-12.

6. Consequences

With s = 4/(x0 - 1) = 1.3 * 10^-13:

• t-free. log(1/zeta(t)) + s + E* < 0 decides t <= 25 for both routes (at t = 25: -2.98035 * 10^-8 + 2.48068 * 10^-8 < 0, margin 5.0 * 10^-9) and fails at t = 26 (-1.49016 * 10^-8). With Lemma 3(c) and Morrill-Platt below x0#: Corollary 1.
• Valuations. Lemma 3(a)-(b) give Robin whenever q^(nu_q(n)+1) < Q*, Q* = 1/(1 - exp(-(E* + s))); each listed exponent is decided against the lower endpoint of the Q* ball. Corollary 2, which contains Corollary 1 via q = 2.
• Unconditional constant. sigma(n)/n < n/phi(n) <= e^gamma log log n exp(E*): Corollary 3 with epsilon = exp(E* + s) - 1.

7. Against the prior method

The first two margins are approximate, read from the papers' final inequalities and not recomputed here; a test checks only that each lies between log zeta(t+1) and log zeta(t) for that paper's t. The theta bounds in the first row are quoted.

Morrill-Platt's v1 (arXiv:1809.10813v1, 2018) claimed 25-free from one computed value R_25(N_177244758016) < 1 - 10^-16 and a monotonicity of R_t(N_n) it did not prove; v4 withdrew to 20 and its section 5 says the monotonicity would have made the proof easy. Corollary 1 recovers 25 by a different argument. Robustness: route B keeps t = 25 if every Table 1 entry is raised by 0.5, and loses it at +0.6; route A loses it if 1.95 is replaced by 2.6 (both pinned).

8. Checks (test_bound.py, 37 tests, ~45 s)

• Lemma 1 on real primes, two ranges, both the sum form and the E difference form; a flipped sign is caught. Lemma 2 at 400 random points.
• Closed-form integrals against mpmath.quad, 12 cases; the whole route-B main term against blockwise quadrature.
• The bound machinery, started at x = 10^5, 10^6, 3 * 10^6 (global 0.94 below 10^10), dominates the true E(x) computed from the primes; the same comparison with the integral scaled by 0.25 fails at some sampled x (the planted fault is caught).
• Inputs: Table 1 covers [x0, 10^19] with no gap and its maximum is Buthe's 0.94; Table 8 ordering; Buthe's 0.05 sqrt(x) < x - theta(x) <= 1.95 sqrt(x) at sampled x <= 3 * 10^6; every far-range deficit envelope is below 2 * 10^-8.
• Lemma 3's elementary steps to 2 * 10^5; Robin directly on 5041 <= n <= 30030.
• Pins: t = 25 passes and 26 fails on both routes; E*, Q*, the valuation list and epsilon for both routes; results.json on disk matches a fresh run.
• Every number this file derives is pinned at the precision it is printed (the test_stated_* tests, added 2026-10-10): the section 5 table, the radius, s, the t = 25, 26, 27 values and margin, the separate-terms cost, every range of the valuation list, Axler's list against the threshold read off it, the deficit constants, and the numbers in "The doors" except the two marked there as unpinned estimates. Quoted inputs are constants of bound.py; section 7's prior margins are consistency-checked only. Planting 1.951 for 1.95, one unit in the last digit of Table 8's first row, or 0.006 for the Table 1 margin turns these pins red. x0 is prime, and 10^(10^13.11485) < x0# (so the verified range ends at Corollary 2).

9. Grade, review, novelty

• Lemmas 1-3 and Claim 1's reduction: ordinary derivation, written and checked by one session; identities exact-checked on real primes. No independent review yet. A reviewer of PR #270 reports re-deriving Lemma 1 and recomputing the route-A main term (2.22602 * 10^-8) and the t = 25, 26 decisions; that is a partial check, not a review.
• Awaiting outside review, named: Lemma 3(b) and (c) (the reduction from E to the t-free and valuation statements), and the tables: Buthe's Table 1 (route B only; route A uses his (1.6)) and BKLNW's Tables 8 and 9, the latter computed assuming RH to height 2.446 * 10^12 by a method whose original statement BKLNW correct (section 3). The transcriptions agree with the arXiv versions (section 3); the tables themselves are taken on their authors' word.
• The evaluation of E*: enclosure-carrying (Arb), with an independent quadrature route agreeing.
• The inputs: published, quoted, not recomputed. Buthe's Table 1 and (1.6) rest on his analytic computation; Morrill-Platt's range on their interval-arithmetic run.
• Composite grade: candidate, ordinary derivation, unreviewed, pending external verification.

Search (2026-10-10)

Searched on 2026-10-10, adding to the first session's three web searches (2026-10-03/04) and its full texts of Axler 2023 and Morrill-Platt v1 and v4:

• Web: six queries: Robin inequality t-free integers, Robin's inequality 21-free integers Axler, Morrill Platt Robin inequality 20-free, Robin inequality k-free integers 2025 2026 arXiv, "Robin's inequality" "25-free", "Robin's inequality" free integers 2026.
• arXiv (API, titles and abstracts containing Robin's inequality, Robin's criterion or colossally abundant, newest first, through arXiv:2609.33794 of 2026-09-27).
• zbMATH Open (API: title "Robin's inequality"; title Robin and inequality; abstract "Robin's inequality" and free; title "free integers" and Robin): every number-theory hit from 2022 on read at review or abstract level, earlier ones by title and the surveys in Axler 2023 and Assani, Chester, Paschal.
• Citations: Semantic Scholar's citing lists for Axler 2023 (arXiv:2110.13478, 7 papers) and Morrill-Platt (arXiv:1809.10813, 5 papers).

Found, as they bear on this claim: 21-free remains the best published t-free result (Axler, Ramanujan J. 61 (2023), zbMATH 1532.11010; reproved by Assani, Chester, Paschal, Integers 26 (2026) A86, whose survey of June 2026 also ends at 21). Morrill-Platt's v1 claim of 25-free (2018) is the only 25-free statement found, and its own later versions and the journal version withdraw to 20. Valuation results: nu_2 <= 19 and others (Hertlein, Integers 18 (2018)), nu_2 <= 20 and the list above (Axler 2023), nu_2 <= 16 (Wu, arXiv:1812.00987, unrefereed), Robin for n not divisible by some prime q <= 1 771 559 (Vega, Ramanujan J. 59 (2022)); a Zenodo preprint (Cavalcante, 2026-07-31) bounds valuations only for a hypothetical least counterexample, a different statement. Unconditional constants: 1 + 5.645 * 10^-7 (Hertlein), 1 + 3.15367 * 10^-7 (Axler); Axler-Nicolas 2023 not read. Of the papers read in full (Axler 2023, Morrill-Platt v1 and v4), none uses the first-order cancellation of Lemma 1; the others were read at abstract or review level only. A claimed proof of RH (arXiv:1605.08273, v15) was found and is not counted.

Delta against the best source found: Axler 2023 proves Robin's inequality for every 21-free n > 5040, for nu_2(n) <= 20 and similar valuations, and with constant 1 + 3.15367 * 10^-7, using the same verified range (Morrill-Platt) and separate bounds on the two first-order terms. This hunt claims, as a candidate pending external verification: 25-free, nu_2(n) <= 24 and the list of Corollary 2, and 1 + 2.49 * 10^-8, from the same verified range plus Buthe's 1.95 sqrt(x) and the cancellation. Not searched: MathSciNet, Google Scholar and non-English venues. Original to this lab; novelty not established; no record is claimed beyond what this search found.

10. Reproduction

.venv/bin/python hunts/robin_tfree/bound.py # writes results.json, < 1 s .venv/bin/python -m pytest -n0 -q hunts/robin_tfree/test_bound.py # 37 tests, ~45 s

Requires python-flint (Arb), mpmath, numpy. No network, no heavy compute.

The doors

This hunt measured a ceiling: the largest t the Solé-Planat reduction reaches with E* bounded as above. That is a candidate at x0 = 29 996 208 012 611, not a proved ceiling of the problem.

Active constraints at the optimum, ranked by shadow price.

1. The verified range x0. The binding constraint. The main term is ~ 1.94 c/(sqrt(x0) log x0), so E* halves when x0 grows about fourfold; each further unit of t costs roughly a factor 4 in x0. Raising x0 means re-running Morrill-Platt's colossally-abundant verification (their v1 reports three weeks on one core for its run to 10^(10^13.099)), linear in the prime range.
2. The constant in x - theta(x) <= c sqrt(x) on [x0, ~10^15]. Shadow price dE*/dc = 1.14 * 10^-8 per unit of c (route A). t = 26 needs E* < 1.49 * 10^-8, that is an effective c near 1.1, while u - theta(u) itself averages about 1 * sqrt(u) (the psi(sqrt u) term). So t = 26 at this x0 needs theta computed exactly on [x0, ~10^15] and is marginal even then; t = 27 (7.45 * 10^-9) looks out of reach at this x0 by any upper bound on E(x0) (heuristic scale E(x0) ~ 2/(sqrt(x0) log x0) = 1.2 * 10^-8: the arithmetic is pinned, the scale is neither measured at x0 nor a theorem).
3. The tail past 10^19, 2.55 * 10^-9, 10% of E*, dominated by BKLNW rows b = 40, 45, 50. Buthe 2016's partial-RH Schoenfeld bound to 2.17 * 10^25 would cut it to about 1.8 * 10^-9 (an estimate, not computed by bound.py and not pinned); not used, so the tail rests on one input family, BKLNW's tables.

Frozen-constant inventory. x0 (input; genuine trade shape: compute cost against about half a bit of t per doubling). 1.95 / Table 1 values (input; tighter values need new zero-based computation, genuine trade). +0.005 table margin and 10^-5 BKLNW inflation (safety; no trade, cost 5.7 * 10^-11 on route B, under 10^-13 for the inflation). 1.5 for psi - theta (safety over 1.4262; cost below 10^-11). Lemma 2's 10^-6 and L >= 31 (sized to x0; no trade). s = 4/(x0 - 1) (no trade). Choice of Table 8 over Buthe 2016 on [10^19, 2.17 * 10^25] (genuine trade: about 7 * 10^-10 of margin, the same unpinned estimate, against one more input).

Information class of each door. Door 1 (larger x0) and door 2 (exact theta past x0) require reading more: new prime or colossally-abundant computation beyond what any current table holds. Door 3 stays inside published data. The first-order cancellation is spent; the remaining analytic slack inside this family is Lemma 2's 4 * 10^-15.

What remains open. These estimates leave Robin's inequality undecided on the multiples of M*. Corollary 2 reduces RH to Robin's inequality on that remaining set; the present bound does not establish it there. The scale for E(x0) discussed above is a heuristic, and this derivation proves no necessary bound on E(p_k) for every k and no impossibility theorem for further work on this route. Any stronger conclusion needs a separate argument covering the remaining integers.