li_arithmetic: a uniform bound on the arithmetic part of lambda_n

Hunt #150 (rh_moonshot), mechanism 3. Built 2026-10-21, written up and its prime-side run made on 2026-10-26 when the owner resumed the hunt in place. Status: constructed and measured; not challenged yet; unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how this hunt ended. Files: li_arith_split.py (the split and its routes), li_arith_prime.py (the prime-side form and the bounds), li_arith_wa.py (the W_a rival), the three run_li_arith_*.py drivers, test_li_arithmetic.py (the pins), artifacts/ (every number below), RUNS.md, this page.

Grades are the hunt's (MISSION.md, top): measured, hardened, derived with its review status, kernel-checked; inferred marks a reading no computation here checks. Nothing here is kernel-checked and no Arb enclosure carries any step (the Cauchy routes use Arb as fast multiprecision and drop the radii). A composite claim takes the grade of its weakest step.

1. Target

Li's criterion: RH holds if and only if lambda_n = n [z^n] log xi(1/(1-z)) >= 0 for every n >= 1 (Li 1997; Bombieri and Lagarias 1999). Bombieri and Lagarias split lambda_n into an archimedean part and an arithmetic part. Target: an inequality on the arithmetic part, proved from the Euler product, whose uniformity in n gives lambda_n >= 0 for every n; or else the exact n-range and term at which every candidate inequality fails.

The target is RH-equivalent by construction (it is Li's criterion rewritten), so the only possible progress is an estimate that proves it. Kill condition 3 of MISSION.md applies to any version that offers the rewrite without the estimate.

2. Construction (derivations self-reviewed, each identity pinned by a test)

The split. With s = 1/(1-z) and the factor (s-1) kept with zeta,

lambda_n = A(n) + T(n), A(n) = n [z^n] log( s pi^{-s/2} Gamma(s/2) ) = 1 - (n/2)(gamma + log 4 pi) + sum_{k=2}^{n} C(n,k) (-1)^k (1 - 2^{-k}) zeta(k), T(n) = n [z^n] log( (s-1) zeta(s) ) = sum_{j=1}^{n} C(n,j) eta_{j-1},

with zeta'/zeta(s) + 1/(s-1) = sum_k eta_k (s-1)^k. T(1) = Euler's gamma, and A(n) grows like (n/2) log n. The DH analogue keeps f(s) whole (no pole, f(1) > 0) with its own archimedean factor (li_arith_split.py docstring).

Routes. A(n): a binomial transform of zeta(k) at the precision its 2^n cancellation needs, and a cancellation-free series with a Hurwitz tail. T(n): Stieltjes constants (small n) and a Cauchy integral of log((s-1) zeta(s)) on |z| = r with Arb's zeta. lambda_n itself: an independent Cauchy route on xi, and the tables of hunt #128 (li_dh_onset).

The prime-side form. The Laguerre generating function gives n [z^n] p^{-ks} = n p^{-k} L_n^{(-1)}(k log p), and with E(x) = psi(x) - x and an integration by parts,

lambda_n = A(n) + 1 - L_n^{(1)}(log 2) - P_n, P_n = int_2^oo (psi(x) - x) L_{n-1}^{(2)}(log x) dx / x^2,

so lambda_n >= 0 is exactly P_n <= M(n) := A(n) + 1 - L_n^{(1)}(log 2). The Euler product enters only through psi. A proof of RH along this mechanism is a proof of that inequality for every n.

3. Results (measured unless stated)

The split, zeta and DH, n <= 5000 (artifacts/split_zeta.json, split_dh.json):

A(n) carries lambda_n for both functions; their arithmetic parts are of the same small size. Two routes for A(n) agree at 30 digits on the twelve spot indices up to 5000 (exactly as printed, except 7.7e-22 at n = 10 for zeta; the test holds them to 1e-20 for n <= 60), and two routes for T(n) agree to 1e-18 for n <= 16 (tests).

W_{1/4} = zeta(s+1/4) zeta(s-1/4), n <= 9000 (artifacts/wa_quarter.json): lambda_n > 0 for n < 8260 and lambda_8260 = -122.32; 215 negative values up to 9000, the least -87409.5 at n = 8977. The Cauchy route at r = 0.95 and at r = 0.93 agree to 2.6e-48 (relative), and the zero-side route (zeta's first 1000 zeros and the smooth density; it never evaluates zeta) gives -122.3158 at n = 8260, a difference of 2.5e-3. Grade of the sign: hardened (two independent float routes). The split of W_{1/4}: T_W(2000) = 41.6, T_W(5000) = 22.8, then T_W(8258) = -52042 and T_W(8260) = -55949 against A_W(8260) = 55827. The failure is in the arithmetic part, abruptly.

The prime side, zeta, x0 = 10^8, n <= 5000 (artifacts/prime_side.json, made 2026-10-26):

• The regularized prime sum converges to T(n) for n <= 6: the integrated form is within 8.1e-4 at X = 10^8 (1.1e-3 at 10^6), and the boundary term improves every n.
• M(n) < 0 for n <= 4, and the measured P_n meets it there (P_1 = -0.884 < M_1 = -0.861). For n >= 5, P_n / M(n) lies in [-0.184, 0.454] (largest at n = 8). P_n stays of size 1 to 10 while M(5000) = 15650.
• The pointwise route fails at every n. B(n) = int beta |e^{-u/2} L_{n-1}^{(2)}(u)| du, with beta = abs E exactly below 10^8 and Schoenfeld's envelope sqrt(x) log^2 x / (8 pi) above, which assumes RH. B(n) > M(n) at every n <= 5000 where M(n) > 0; the least ratio is 11.5, the ratio at n = 5000 is 1.57e4, and B(n) / abs(P_n) has median 1.07e7. Even an RH-strength pointwise input does not give the inequality: the true P_n is small by cancellation of the kernel's oscillation against E, not by the size of E.
• The mean-square route fails outside [32, 1258]. sqrt(D2 K2) + tail, with the exact mean square below 10^8, Cramer's RH mean square beyond (a model) and the Schoenfeld envelope past the kernel's turning point, is at most M(n) for n in [32, 1258] (1227 indices) and above it for n <= 31 and every n >= 1259; at n = 5000 it is 3.05 M(n), driven by the tail (3.99e4 against sqrt(D2 K2) = 7.7e3).
• An envelope of de la Vallee Poussin shape x exp(-sqrt(log x / R)), R = 5.573412, prefactor 1 (not a theorem's constants), exceeds M(n) at every n <= 200 and grows like exp(2n) (2.3e18 at n = 10).

All three bounds are trapezoid integrals on a sampled grid: measured, adequate for the factor-of-many comparisons they are used for, not for a sign at a margin.

4. The distinguishing step and the rivals

The step that uses the Euler product is P_n <= M(n) through psi: the inequality must use that the log-derivative coefficients are Lambda(m), supported on prime powers, and it must fail for any function lacking what zeta has.

So: no bound on T(n) or P_n read off n <= 5000 separates zeta from DH (both pass any bound abs T(n) <= 0.012 (n/2) log n on that range: a shared finite-range lemma, no verdict); and W_{1/4} shows that the Euler product with nonnegative coefficients and a functional equation is not enough. What W_{1/4} lacks is that its local roots p^{+-1/4} are not unitary, the same input ccm_arithmetic isolated (its section 4).

5. Known obstructions engaged

• A finite range is a finite height. One off-line quadruple 1/2 +- delta +- i gamma contributes about -2 R^n cos(n psi) to lambda_n with log R ~ delta / ((1/2 + delta)^2 + gamma^2), so it first beats the baseline (n/2) log n near the root of n log R = log((n/4) log n). That root is 7821 for W_{1/4}'s first quadruple (measured onset 8260) and 330025 for DH's off-line pair (extrapolated onset 328997); the test pins both. Inferred (a heuristic that ignores the cosine and the other zeros), but it says what the record already implies: positivity of lambda_n for n <= N says nothing about off-line zeros above a height of order sqrt(N delta / log N), so any check to n = 5000 is a check to bounded height, and a uniform-in-n step is the whole content (docs/08 section 3).
• The pointwise and mean-square routes are absolute-value routes, and on the measured range both fail even with RH-conditional inputs (section 3), which also makes them circular as written. A proof has to exploit the sign structure of E against the Laguerre kernel.
• The zeros route to lambda_n is structurally nonnegative over any list of on-line zeros (MISSION.md dead routes); nothing here uses it as evidence for zeta. W_{1/4}'s zero-side route is a cross-check of a negative value, not positivity evidence.

6. Tests run

test_li_arithmetic.py, 15 tests, 21 to 27 s with -n0, writing nothing into the tree: closed forms at n = 1; the two A routes and the two T routes for zeta and DH; a planted fault (the Euler factor of 2 removed) that the T cross-check must see; the identity against the hunt #128 tables; the Laguerre generating identity and the elementary piece on [1, 2); the regularized prime sum against T(n) at X = 10^6; the pointwise route recomputed at x0 = 10^6, n <= 40; the W_{1/4} split at small n by three routes; the radius rule; the onset heuristic; and every artifact number this page quotes.

7. Disposition (proposed by the builder; the challengers decide)

Unresolved, with two sub-routes refuted at their stated parameters: the pointwise absolute-value bound (at every n <= 5000 where M(n) > 0, with x0 = 10^8 and the Schoenfeld envelope above it) and the mean-square bound (at n <= 31 and every 1259 <= n <= 5000, with Cramer's mean square as a model above x0). Both refutations are of the bounds, not of the inequality P_n <= M(n), which holds on the whole measured range.

8. Furthest true statement, with its grade

For 1 <= n <= 5000, lambda_n = A(n) + T(n) holds to 8.1e-28 for zeta and 3.1e-27 for DH, abs T(n) <= 13.6 for zeta and <= 19.2 for DH while A(n) reaches 1.6e4 and 2.0e4, the prime integral satisfies P_n <= M(n) with P_n / M(n) <= 0.46 for n >= 5, and no bound on P_n by abs E (pointwise, even granting Schoenfeld's RH envelope) reaches M(n) at any of those n; W_{1/4}, which has an Euler product, nonnegative log-derivative coefficients and a functional equation, has lambda_8260 < 0 with the failure in its arithmetic part. Grade: measured, except W_{1/4}'s sign (hardened, two routes) and the split's derivation (derived, self-reviewed). No novelty search was made: the split is Bombieri-Lagarias's and Coffey's, and the prime-side Laguerre form is the integrated version of Bombieri and Lagarias's arithmetic formula; whether the measured failure of the absolute-value routes is in the literature was not searched.

9. Next step

1. A signed bound: write P_n against the explicit formula for E and bound the sum over zeros of the Laguerre transform zero by zero; that reads the zeros and so is a different information class from the prime-side input (MISSION.md, zero-importing). The arithmetic alternative is a large-sieve or mean-value bound on sum_m Lambda(m) m^{-1/2} e^{i t log m} weighted by the kernel's phase 2 sqrt(n u).
2. Run Epstein (1,1,6) and zeta(s - delta) through the split, to see which hypothesis of the inequality each lacks.
3. Compute DH's lambda_n to the predicted onset only if a candidate inequality survives to n = 5000 for zeta and its DH analogue does not yet fail there; at today's cost (150 s for n = 5000 at r = 0.95, with work digits growing like n log10(1/r)) the run to n = 3.3e5 needs its own estimate in RUNS.md first.