Claims · 18 dated claims on The digit-restricted Mobius exponent, each with its tag and its witness; newest 2026-09-23.
Conjecturetheta(F) = 1/2 for every digit set with 2 <= abs(F) <= base - 1 and squarefree digit gcd - square-root cancellation against the set's own counting function: the 47 running-maximum exponents across base = 3, 4, 5, 10 read 0.4465..0.5358 with last-five-level drifts 0.0157..0.1056, while the full-set controls, whose limiting exponent is 1/2 under RH and at least 1/2 unconditionally, read 0.4413..0.4517 at the same depths; the finite tables are consistent and decide nothing, single-cut exponents scattering 0.22..0.53 on the same data. Witness: lab/rs/mobius-designs, mobius.md.
Conjecture An unconditional Mertens-shape bound on the dense column: for F omitting exactly one digit, base >= 92317 and x = base^level with level past a point depending on base alone, abs(M_F(x)) <= C(base) A_F(x) exp(-c(base) sqrt(log x)) with C(base) and c(base) > 0 effective, through a Dirichlet-approximation dissection whose region A is the ladder rung b = 4/5 and whose one load-bearing minor-arc input is unread. Witness: lab/rs/mertens-numerology for the wall 92317, Maynard 2022 for the imported lemmas.
Proved That dissection is dead at fixed digit count: {0,1} at base 3 has l^1 exponent log 2 / log 3 = 0.630929, above every bar the dissection sets, region B's 1/4 and region A's own ask included, the l^1 floor again in arc-local form. Witness: exact arithmetic in the sentence that prints it.
Proved Vaughan's identity is circular here at power strength: the mu_{<=U} * mu_{<=U} * 1 piece carries the main term fill^level M_1(U)^2, so bounding the pieces one by one at power strength forces M_1(U) << U^(-delta), which continues 1/zeta into sigma > 1 - delta. Witness: the identity, carried out in the sentence that prints it.
Verified The minor-arc bound in d-form for mu, the one load-bearing analytic input of the unconditional dissection on mobius.md, is read at source: Basak, Robles and Zaharescu 2023, Theorem 1.4, S_mu(theta) <<_eps x^(4/5 + eps) + x d^(-1/2) (log x)^3 + (x d)^(1/2) (log x)^3 on abs(theta - l/d) <= 1/d^2, (l, d) = 1, x >= 2; the first term carries an eps the page's x^(4/5) did not, and region A's strict inequality c_base + 4/5 < alpha_base absorbs it, so the wall 92317 stands. Witness: arXiv:2312.17435 Theorem 1.4 and its proof, Section 3.1; mobius.md the unconditional dissection.
Proved An unconditional Mertens-shape bound on the dense column: for F omitting m digits under the wall condition (W), PB_base(m) < fill base^(-4/5) or at m = 1 and base >= 36 its chord form PB'_base(1, e_0) < (base - 1) base^(-4/5), abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)) at every x >= 2, with C and c > 0 effective and depending on base and F alone; at one excluded digit (W) holds at every base >= 39363, and at every base >= 28352 when e_0 is 0 or base - 1. The four bookkeeping items are written: the Perron constants of region C2, through Koukoulopoulos 2019 Theorem 7.2 and Exercise 8.4 and Chang and Martin 2019 Lemmas 2.1 and 2.4 and Proposition 2.3, the exceptional character real by conjugation and confined to a finite family fixed by base; x off the powers of the base, by a split into at most fill + 1 blocks per scale with the lowest kappa sqrt(log x) scales discarded; the region C1 transfer, the perturbed Lemma A' at dim = 1 being stated at the grid point itself; and every m, (W) forcing two consecutive digits. The shape is in print for the von Mangoldt function, Maynard 2022 Theorem 1.1 at base > 2 * 10^6, where a Mobius version is a routine unwritten adaptation; the new content is the object mu with an effective constant and the wall 39363. Witness: mobius.md The unconditional dissection; lab/py/mobius-dissection, verbs wall, blocks, regions.
Verified The wall of that theorem at one excluded digit, the ladder rung b = 4/5 read with the chord constant: 39363 over every excluded digit and 28352 at e_0 in {0, base - 1}, each an up-set of the float scan 36..2 * 10^5 (held 160638 and 171649), the gap (base - 1) base^(-4/5) - PB'_base(1, e_0) at 40 digits <= -2.3195 * 10^-6 at 39362 and >= 2.3677 * 10^-5 at 39363, <= -1.3539 * 10^-5 at 28351 and >= 1.8837 * 10^-5 at 28352; the step 3 constant closes at 92317, and above the scan the chord bound's PB'_base(1, e_0) < PB_base(1) and the proved floor of the ladder make both walls half lines. Witness: lab/py/mobius-dissection, verb wall.
Proved A hybrid l^1 bound from the one-shift constant alone: for y = base^k and 16 base^2 D (D + H) <= y, the residues a mod y whose Dirichlet fraction at level y^(3/5) has D <= d < 2D and height h < 2H carry sum abs(hat F_k(a/y)) <= (1 + pi base) fill^k (V_1 V_2)^(alpha_1) with V_1 V_2 < 16 base^2 D (D + H) and fill^i base^(i alpha_1) any proved bound on the shifted-grid sum at base^i points, by a three-block split of the transform, Gallagher's inequality on Farey intervals disjoint mod 1, and norm(f')_1 <= pi base B_base(F)^(i_1) from the l^1 floor B_base(F) >= base; no large sieve is quoted. Witness: mobius.md The unconditional dissection, the hybrid bound bullet.
Proved Region B of the dissection asks only alpha_1 < 1/4, with the block depth kappa log base < C_0 (1/2 - 2 alpha_1), and that is the GRH certificate PB < fill base^(-3/4) of the rung b = 3/4, closing from 1499 at one excluded digit; region A, which pays the whole l^1 mass against x^(4/5), alone sets the wall. Witness: mobius.md The unconditional dissection, region B.
Proved At one excluded digit no constant bounding the shifted-grid supremum B_base(F) brings the dissection's wall below base 33: the floor B_base(F) >= 2(base - 1) makes region A ask (base - 1) base^(-4/5) > 2 - 2/base, that is base^(1/5) > 2; the unshifted mass c_k region A pays has only Parseval's floor base^k proved, and its ratios c_4/c_3 = 74.1654 at base 33 missing 16 and 70.5661 missing 0 against 64 are readings. Witness: mobius.md The unconditional dissection, what lowers the wall; lab/py/mobius-dissection, verb wall.
Verified The dissection's bookkeeping on the whole grid of base 10 missing 5 and missing 0 at level 6 and base 5 missing 2 at level 9: the four regions sum to the exact sum of mu over the strings, C1 and C2 stay under 3 Z^2 and 3 Z N_Z, every C2 denominator divides y, the second approximation lands in [y/(2h), 2y/h] at all 74778 and 128942 points of region B with h >= 1, and the hybrid bound and its Gallagher step hold at every class meeting the two conditions the proof uses, 73 at each base-10 design and 86 at base 5, the largest ratio past the class of a = 0 at most 0.001356 and the Gallagher step, both norms read on a grid, at most 0.110814; the perturbed Lemma A' bound holds at 6000 seeded grid points of level 30 in base 10 and level 45 in base 5 with logarithmic margin at least 27.9; the block split meets M_F(x) exactly at 400 random x below 2 * 10^6 and at every base^e - 1 in five families. Witness: lab/py/mobius-dissection, verbs regions and blocks.
Verified Koukoulopoulos 2019 Theorems 23.5 and 23.6, the Type I and Type II estimates behind Basak, Robles and Zaharescu 2023 Theorem 1.4, read at source in the author's preliminary text, pp. 242 and 245: sum_(n <= x) (f * log^v)(n) e(n alpha) << (y + x/q + q)(log x)^(v+1) norm(f)_inf at every v >= 0, and sum_(n <= x) (f * g)(n) e(alpha n) << (q + y + z + yz/q)^(1/2) sqrt(log 2q) norm(f)_2 norm(g)_2, both with absolute constants; the restatement in Basak, Robles and Zaharescu 2023 Lemma 2.1 reads v >= 2 and is applied at v = 0, which the book's v >= 0 covers, and the proof of their Theorem 1.4 uses no ineffective input, their one Siegel-Walfisz lemma sitting in the major-arc section. Witness: https://dms.umontreal.ca/~koukoulo/documents/publications/primes.pdf, arXiv:2312.17435v2.
Proved The unconditional Mertens-shape bound restated on its l^1 input: if F keeps two consecutive digits and carries a shifted-grid certificate below 1/5, constants C_F >= 1 and alpha_1 < 1/5 with sum_(a < base^i) abs(hat F_i(s + a/base^i)) <= C_F fill^i base^(i alpha_1) at every i >= 0 and every real s, then abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)) at every x >= 2, C and c > 0 effective; region A reads the certificate at s = 0, the unshifted mass, region B through the hybrid bound at every shift. Every one-missing-digit set from base 584 carries such a certificate by the digit-uniform chain of coprime.md, so the bound holds there, and it holds at every one-missing-digit set of base 301..583 on the window certificates of the same subsection; the wall condition (W) is a certificate with C_F = 1, and alone it closes at 39363, and at 28352 when the excluded digit is 0 or base - 1. Witness: mobius.md The unconditional dissection; coprime.md The dissection for the von Mangoldt function; lab/py/prime-dissection, verbs wall and window.
Proved The hybrid l^1 bound on a shifted-grid certificate: under 16 base^2 D (D + H) <= y, the residues a mod y whose Dirichlet fraction at level y^(3/5) has D <= d < 2D and height h < 2H carry sum abs(hat F_k(a/y)) <= C_F^2 (1 + pi base^2 C_F/fill) fill^k (V_1 V_2)^(alpha_1), with norm(f)_1 from the certificate at every shift integrated and norm(f')_1 <= (pi base^2/fill) C_F^2 fill^(i_1) V_1^(alpha_1) from the product with one factor removed, split into base shifted grids; at C_F = 1 with the per-digit constant it is (1 + pi base) fill^k (V_1 V_2)^(alpha_1). Witness: mobius.md The unconditional dissection, the hybrid bound bullet.
Verified The falsification of the dissection's hybrid bound, counted as its generator prints it: the C_F = 1 form holds at the 73, 73 and 86 classes meeting V_1 V_2 <= y and 16 D H <= y at base 10 missing 5 and missing 0 at level 6 and base 5 missing 2 at level 9, largest ratio past the class of a = 0 at most 0.001356, and the Gallagher step of its proof at the 71, 71 and 75 of those classes other than that of a = 0 with V_1 <= 10^5, largest ratio at most 0.110814 with both norms read on 4 V_1 points. Witness: lab/py/mobius-dissection, verb regions.
Verified The unconditional Mertens-shape bound abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)), C and c > 0 effective, holds at every one-missing-digit set from base 115: every such set of base 115..300 carries a shifted-grid certificate alpha_1 < 1/5 by per-digit window certificates, of base 301..583 by the digit-uniform window, and from base 584 by the chain, where the bound is Proved; 115 is the floor of the route at one missing digit, every base 3..114 carrying a missing digit certified alpha_1 > 1/5, base 114 missing 56 among them, while single sets clear from base 65 missing 0. Witness: mobius.md The unconditional dissection; coprime.md The least base below a fifth is 115; lab/py/digit-transform-norms, verbs fifth and fifthbelow.
Proved The unconditional Mertens-shape bound on the dense column, narrowed to what is proved: abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)), C and c > 0 effective, at every F that keeps two consecutive digits and carries a shifted-grid l^1 certificate below 1/5, at every F under the wall condition (W), and at every one-missing-digit set from base 584 by the digit-uniform chain; its reach at one-missing-digit sets of base 115..583 rests on window certificates, and the location of the (W) walls on a float scan, and both are Verified and not Proved. Witness: mobius.md The unconditional dissection; coprime.md The dissection for the von Mangoldt function.
Verified The (W) walls at one excluded digit are float-scanned and not interval-certified: 39363 over every excluded digit and 28352 at e_0 in {0, base - 1} are the least bases of an exhaustive float64 scan of 36..2 * 10^5, each an up-set of it, with the gap re-read at 40 digits at each wall and one below, and above the scan the proved PB'_base(1, e_0) < PB_base(1) and the step 3 wall 92317, itself scanned, carry them; the rows that place these walls under a proved statement read them at this tag. Witness: lab/py/mobius-dissection, verb wall; mobius.md The unconditional dissection, what the per-digit constant alone gives.