
The Mobius meter across digit designs
Fix a base q >= 3 and a digit set F inside {0..q-1} with k = |F| >= 2. The digit-restricted set S_F holds the positive integers whose base-q digits all lie in F, with no leading zero: a one-dimensional digit design, the same restriction rule that carves every fractal in this tree, read on the integer line instead of the square. This page measures how much the Mobius function cancels along each design, against the design's own size - and proves that the columns are not independent: digit sets that are scalar multiples of each other carry exactly transferred meters, including one family whose meter vanishes identically and one base-4 pair locked in exact anti-symmetry. Every number is printed by lab/mobius-designs, the divisor section's by lab/rho-decoupling, and the GRH section's by lab/mertens-numerology.
Tags as everywhere in this tree: Proved means derived here from definitions, Verified means recomputed exactly and checked against an independent path, Conjecture is labelled belief.
The zeros are the other face and they are a different page. The design's own Dirichlet series zeta_F(s) = sum_(n in S_F) n^(-s) has zeros inside its own half-plane of absolute convergence, a comb of them along the pole lattice, three products where the integers have one Euler product, and a Mertens function of its own that runs the wrong way; none of it reaches the meter measured here, and the decoupling is why this page's question is about mu restricted to S_F and nothing else: zeta.
The meter and its yardstick
A_F(x)countsS_Fup tox. The count is exact at every checkpoint:A_F(q^L) = k^L - 1when0 in F(plus 1 when1 in Ftoo, for the boundary elementq^Litself), andA_F(q^L) = (k^(L+1) - k)/(k - 1)when0is not inF, by counting digit strings of each length. Proved; the lane's tests pin it against direct enumeration. Between checkpointsA_F(x)/x^(log_q k)carries the log-periodic ripple every design in this tree carries - the classical fluctuation of digital sums (Flajolet, Grabner, Kirschenhofer, Prodinger and Tichy 1994) - so a checkpoint value is a grid value, never a constant.- The meter is
M_F(x) = sum of mu(n)overn in S_F,n <= x, and the exponent istheta(F) = limsup of log|M_F(x)| / log A_F(x). A single cut of|M_F|is a bad estimator - the meter crosses zero freely - so the census prints two readings per level:M_F(q^L)itself, and the running maximummax of |M_F(x)|overx <= q^L, whose exponentthetamaxis monotone in the numerator and is the estimator the slope tables use. - The yardstick matters.
A_F(x)grows likex^(log_q k), soS_Fis sparse, and a bound of shapeo(x)is weaker than the trivial|M_F(x)| <= A_F(x). The indicator ofS_Fis aq-automatic sequence, so Mullner 2017 (automatic sequences fulfill the Sarnak conjecture) givesM_F(x) = o(x)for everyF: orthogonality holds and the question is well-posed, but against the set's own mass that bound says nothing at all. The same shape repeats in base 2 through circuits: the indicator is computable in bounded depth from the binary digits, so Green 2012 also giveso(x), again below the trivial bound. Verified against the literature. The honest question istheta, and it is open at every2 <= k <= q - 1. - The Dirichlet series over
S_Fis built territory, and this page claims nothing about it: the abscissa islog_q k(Kohler and Spilker 2009, with position-varying digit rules in Nathanson 2021); the series continues meromorphically toCwith simple poles amongs = log_q k - m + 2 pi i j / log q(the automatic-series mechanism of Allouche, Mendes France and Peyriere 2000, carried out for missing digits in Burnol 2026 and unified in Allouche, Shallit and Stipulanti 2025); a pole lattice of period2 pi i / log qreads as log-periodic oscillation through the Mellin dictionary of Flajolet, Gourdon and Dumas 1994, and the oscillation is visible in the series' own numerical moments (Burnol 2026 oscillations); the Mobius function itself is notk-automatic for anyk, so the Mobius-weighted series inherits none of that continuation (Coons 2010); and no Mobius or Mertens sum appears anywhere in that literature. Verified against the sources in REFS.md. The series does not carry the meter the way zeta carries Mertens:S_Fis not multiplicatively closed - atq = 3,F = {0,1}, both4 = 11and13 = 111lie inS_Fwhile4 x 13 = 52 = 1221does not - so there is no Euler product andM_Fis not the coefficient sum of an inverse series. Proved by that witness. - The full digit set is the classical boundary.
S_Fis then every integer,M_Fis the Mertens function of Mertens 1897, andM(x) = O(x^(1/2 + eps))for everyeps > 0is equivalent to the Riemann hypothesis (Titchmarsh 1986, Theorem 14.25 (C)), whilelimsup |M(x)|/sqrt(x) >= 1.06unconditionally by Odlyzko and te Riele 1985, so the exponent over allxequals1/2exactly when RH holds. Verified against the literature. This page claims nothing about RH: the full-set column below is a control rendered for scale, and the tree's own claims live in the restricted columns. - The even moments of the digit transform are additive energies. Proved.
sum_{a mod q^L} |hat F_L(a/q^L)|^(2r) = q^L E_r(L)withE_r(L)the number of2r-tuples of length-Ldigit strings withn_1 + ... + n_r = n_(r+1) + ... + n_(2r) mod q^L, by orthogonality, andE_r(L)is counted by a carry DP on the carry pairs of the two sides, so each moment is C-finite inLof order at mostr(r+1)/2and its growth constantLambda(2r) = q rhois an algebraic number,rhothe Perron root of the transfer matrix, certified in exact rationals (lab/rho-decoupling, therieszmodule). - The fourth-moment constants. Verified.
Lambda(4) = 18at{0,1},{0,2}and{1,2}in base 3 (rho = 6),2(23 + sqrt 353) = 83.5766at{0,1,2}in base 4 (x^2 - 23x + 44),(275 + 5 sqrt 2369)/2 = 259.1809at{0,1,2,3}in base 5 (x^2 - 55x + 164),95at{0,2,4}in base 5, and6566.412to6567.410over the reflection classes of one excluded digit in base 10; every value sits strictly inside[max(k^4, q k^2), q k^3]and a hair abovek^4at the dense families (log_q(Lambda(4)/k^4)is0.107at base 3,0.0004at base 10), and the sixth, eighth and tenth moments at{0,1}base 3 are39 + 3 sqrt 79,3(99 + sqrt 5265)/2and a cubic (lab/rho-decoupling). - What a moment buys the bilinear sum. Proved. Holder with the
2r-th moment on the digit side and Parseval on the bilinear side bounds the Type II sum overmup toMandlup toN,4MN <= x, byx^(theta_p/p + 1/2 - 1/p)withtheta_p = log Lambda(p)/log q, which is at leastx^(alpha + 1/4)for every evenp >= 4and every digit set, above the trivialx^alpha; so no moment of the digit transform alone beats the trivial bound, and the route needs the bilinear sum on the minor arcs below its own root mean square, which random-sign coefficients defeat on the census (Verified,lab/rho-decouplingthearcslines). - The multiplicative energy of a digit column has no exponent of its own. Proved. With
E_x(L) = #{(n_1, n_2, n_3, n_4) in D_L^4 : n_1 n_2 = n_3 n_4}andK = k^L, the two diagonals give2K^2 - K <= E_x(L), andE_x(L) = sum_m r(m)^2 <= K^2 max_m r(m)withr(m) <= d(m)givesE_x(L) = k^(2L) x^(o(1))for every base and digit set (the census reads58760487at{0,1}base 3,L = 12, the exponent1.356938falling toward2 alpha = 1.261860); so a Type II sum estimated through the energy obeys|Sigma| <= (2MN)^(1/2) x^(alpha/2 + o(1))and misses the trivial bound by(1 - alpha)/2; the excess over the diagonal is structure, not arithmetic: the shift family(q^i u, q^j v, q^(i') u, q^(j') v)withi + j = i' + j', counted in closed form when0is a digit, is0.44of it at{0,1}base 3,L = 12(lab/rho-decoupling, themenergymodule). - The Mobius signs cancel the column's correlation no better than random signs. Verified. The digit column carries a real off-diagonal multiplicative correlation, zero in the mean for a random column of the same density, and the Mobius and Liouville signs cancel it no better than an unstructured sign vector on the same support does,
|Sigma_mu|sitting at0.0913to0.7178of the random-sign root mean square against0.0359to1.5048for the support-matched controls over sixteen boxes, with the split against those controls3, 9, 4at chi-square0.375against the uniform-rank null; a sign vector built by greedy flips against a known column drives the same Cauchy-Schwarz bound to0.0265of its diagonal floor, so the census measures the arithmetic of the coefficients and not a limit of the method (lab/rho-decoupling,menergy signedandmenergy signed engineered). - The coefficient the method is given is not the coefficient it would need. Verified. At seven of the eight swept boxes with both sides above
x^(2/5), the boxes a Vaughan decomposition actually produces, the coefficient sequence it hands the bilinear sum takes values in{-1, 0, 1}and needs no normalisation, and its full quadratic form sits between0.69and1.21of its own diagonal, where a sign vector engineered against the column reads0.13to0.21on the same boxes, so the sequence the method is given and the sequence the method would need are different objects (lab/rho-decoupling,menergy signed vaughan). - The large-values refinement is the moment route itself. Verified. The large-values refinement of the moment route is costed out and is the
l^2route itself, exponent(1 + alpha)/2at every threshold (lab/rho-decoupling,riesz large values chain, 66 cells over six families); the large frequencies are adjacent grid points (407in331runs at{0,1}base 3,L = 12,eta = eta_4), so the grid offers no spacing gain, and at the dense families the bilinear sum ata_m = b_l = 1equals the box representation count,0.38 k^Lthere, so no bound uniform over bounded coefficients holds at those families.
The exact transfer between designs
The census columns are tied together by one carry-free mechanism. Proved:
- Scaling. If every digit of
Fisatimes a digit ofF', soF = aF'inside{0..q-1}, thenm -> ammapsS_F'bijectively ontoS_Fpreserving digit length:am = sum (a d_j) q^jand eacha d_j <= q - 1, so no carry occurs and the digit string scales digitwise. HenceA_F(q^L)equals the string count ofF'at the same depth, andM_F(q^L) = sum of mu(am)overm in S_F'with at mostLdigits. - Vanishing. If
ahas a square factor thenmu(am) = 0for everym, soM_Fis identically zero: atq = 5,F = {0,4} = 4 x {0,1}, the meter reads 0 at all 21 levels. A census that reads cancellation without factoring out the digit gcd reads this as infinite cancellation; the digit gcd must be squarefree beforethetameans anything. - Prime twist. If
a = pis prime thenmu(pm)is-mu(m)onp-freemand0otherwise, soM_(pF')(q^L) = -sum of mu(m)over them in S_F'not divisible byp. Atq = 3,F' = {0,1}: an elementm = sum of 3^jis odd exactly when its count of 1-digits is odd, and reading the digit string as a binary index that parity is the Thue-Morse sign, so the{0,2}column is the Thue-Morse-twisted{0,1}column. - Base-4 anti-symmetry. At
q = 4,M_{0,2}(4^L) = -M_{0,1}(4^L)exactly: since4 | q, an element ofS_{0,1}is0or1 mod 4by its unit digit, so every even element is divisible by 4 and carriesmu = 0, and the odd-part twist above is minus the whole meter. Stronger,M_{0,2}(x) = -M_{0,1}(x/2)at every realx, and sinceS_{0,1}has no element strictly between(4^L - 1)/3and4^Lthe running maxima agree level by level as well. The census confirms both at all 22 levels, e.g. meters-110/110atL = 15,-342/342atL = 17,34/-34atL = 22, andMmax = 1553for both atL = 22.
Verified: the generator recomputes all eight scaled census families ({0,2} at q = 3; {0,2}, {0,3} at q = 4; {0,2}, {0,3}, {0,4}, {2,4}, {0,2,4} at q = 5) from their primitive families through mu(am) and asserts equality at every level. The mechanism needs a common digit factor, so it partitions the census into primitive columns and their twists and says nothing across primitive columns.
The census
Every M_F(q^L) below is an exact integer: restricted families enumerated in ascending order with mu from deterministic factorization (trial division, Miller-Rabin on the twelve witnesses 2..37, Pollard rho), controls by a linear Mobius sieve; one family (q = 3, F = {1,2}, L = 16) is computed by both methods and asserted equal at every level. The base-10 control reproduces A084237 (-1, 1, 2, -23, -48, 212, 1037, 1928 at 10^1..10^8). Every table below is extracted by script from the generator's printed rows, never assembled by hand. Verified.
The three base-3 columns, checkpoint meter and running maximum Mmax = max of |M_F(x)| over x <= 3^L per row:
L | M_{0,1} | max | M_{0,2} | max | M_{1,2} | max |
|---|---|---|---|---|---|---|
| 4 | -2 | 3 | 2 | 3 | -8 | 8 |
| 6 | 2 | 5 | 0 | 3 | -8 | 11 |
| 8 | 2 | 8 | 3 | 7 | -31 | 33 |
| 10 | 5 | 13 | 0 | 11 | -14 | 38 |
| 12 | 56 | 61 | -37 | 40 | -35 | 88 |
| 14 | 11 | 105 | -10 | 67 | -205 | 230 |
| 16 | 149 | 173 | -124 | 152 | 4 | 281 |
| 18 | -30 | 312 | 67 | 249 | -1461 | 1582 |
| 20 | 496 | 539 | -382 | 485 | -3175 | 3255 |
| 21 | 533 | 866 | -194 | 617 | -2005 | 3855 |
| 22 | 1009 | 1089 | -1205 | 1324 | -690 | 3855 |
| 23 | 1824 | 2848 | -2242 | 2942 | -3214 | 3855 |
| 24 | -1886 | 3296 | -133 | 3843 | -3248 | 4113 |
The final checkpoint of every family, with thetamax = log(Mmax)/log A and its drift (max minus min) over the last five levels:
q | F | L | A_F(q^L) | M_F(q^L) | Mmax | thetamax | drift |
|---|---|---|---|---|---|---|---|
| 3 | 01 | 24 | 16777216 | -1886 | 3296 | 0.4869 | 0.0452 |
| 3 | 02 | 24 | 16777215 | -133 | 3843 | 0.4962 | 0.0596 |
| 3 | 12 | 24 | 33554430 | -3248 | 4113 | 0.4802 | 0.0754 |
| 4 | 01 | 22 | 4194304 | 34 | 1553 | 0.4819 | 0.0391 |
| 4 | 02 | 22 | 4194303 | -34 | 1553 | 0.4819 | 0.0391 |
| 4 | 03 | 22 | 4194303 | -541 | 1180 | 0.4638 | 0.0488 |
| 4 | 12 | 22 | 8388606 | -855 | 3965 | 0.5197 | 0.1056 |
| 4 | 13 | 22 | 8388606 | -712 | 2631 | 0.4940 | 0.0727 |
| 4 | 23 | 22 | 8388606 | -3255 | 3258 | 0.5074 | 0.0157 |
| 4 | 012 | 14 | 4782969 | -503 | 1057 | 0.4527 | 0.0475 |
| 4 | 013 | 14 | 4782969 | 2313 | 2899 | 0.5183 | 0.0487 |
| 4 | 023 | 14 | 4782968 | -753 | 1166 | 0.4591 | 0.0862 |
| 4 | 123 | 14 | 7174452 | -592 | 1644 | 0.4691 | 0.0729 |
| 5 | 01 | 21 | 2097152 | 153 | 849 | 0.4633 | 0.0485 |
| 5 | 02 | 21 | 2097151 | 250 | 889 | 0.4665 | 0.0268 |
| 5 | 03 | 21 | 2097151 | -116 | 700 | 0.4501 | 0.0311 |
| 5 | 04 | 21 | 2097151 | 0 | 0 | - | - |
| 5 | 12 | 21 | 4194302 | -128 | 1643 | 0.4856 | 0.0732 |
| 5 | 13 | 21 | 4194302 | -2875 | 3533 | 0.5358 | 0.0456 |
| 5 | 14 | 21 | 4194302 | -1511 | 2750 | 0.5193 | 0.0640 |
| 5 | 23 | 21 | 4194302 | 405 | 914 | 0.4471 | 0.0581 |
| 5 | 24 | 21 | 4194302 | 1065 | 2287 | 0.5072 | 0.0540 |
| 5 | 34 | 21 | 4194302 | -2137 | 2538 | 0.5141 | 0.0401 |
| 5 | 012 | 13 | 1594323 | -1016 | 1416 | 0.5080 | 0.0846 |
| 5 | 013 | 13 | 1594323 | -137 | 768 | 0.4652 | 0.0528 |
| 5 | 014 | 13 | 1594323 | 213 | 1005 | 0.4840 | 0.0821 |
| 5 | 023 | 13 | 1594322 | 759 | 858 | 0.4729 | 0.0455 |
| 5 | 024 | 13 | 1594322 | 686 | 959 | 0.4807 | 0.0240 |
| 5 | 034 | 13 | 1594322 | 501 | 1000 | 0.4837 | 0.0847 |
| 5 | 123 | 13 | 2391483 | 88 | 816 | 0.4565 | 0.0489 |
| 5 | 124 | 13 | 2391483 | 1036 | 1613 | 0.5029 | 0.0604 |
| 5 | 134 | 13 | 2391483 | -725 | 981 | 0.4690 | 0.0519 |
| 5 | 234 | 13 | 2391483 | -1926 | 2021 | 0.5182 | 0.1056 |
| 5 | 0123 | 11 | 4194304 | -474 | 1725 | 0.4887 | 0.0732 |
| 5 | 0124 | 11 | 4194304 | 426 | 1494 | 0.4793 | 0.0673 |
| 5 | 0134 | 11 | 4194304 | -644 | 1633 | 0.4852 | 0.0222 |
| 5 | 0234 | 11 | 4194303 | -362 | 2179 | 0.5041 | 0.0794 |
| 5 | 1234 | 11 | 5592404 | 145 | 1101 | 0.4508 | 0.0996 |
Base 10 with one digit excluded, the Kempner designs (k = 9, the sets behind the convergent harmonic series of Kempner 1914, revisited at s = 1 in Allouche, Hu and Morin 2024), at x = 10^8:
| excluded | A_F(10^8) | M_F(10^8) | Mmax | thetamax |
|---|---|---|---|---|
| 0 | 48427560 | 6410 | 8177 | 0.5091 |
| 1 | 43046720 | 4108 | 6069 | 0.4956 |
| 2 | 43046721 | -183 | 3357 | 0.4619 |
| 3 | 43046721 | 455 | 3512 | 0.4644 |
| 4 | 43046721 | 56 | 4957 | 0.4841 |
| 5 | 43046721 | -7614 | 10601 | 0.5273 |
| 6 | 43046721 | -693 | 2564 | 0.4465 |
| 7 | 43046721 | -1411 | 6494 | 0.4994 |
| 8 | 43046721 | 2131 | 4495 | 0.4785 |
| 9 | 43046721 | 2181 | 5234 | 0.4871 |
The full-set controls at comparable depth: M(3^17) = -1423 with Mmax = 4610 (thetamax 0.4517), M(4^13) = 329 with 2845 (0.4413), M(5^11) = 617 with 2573 (0.4436), M(10^8) = 1928 with 3448 (0.4422). The Mertens function itself - limiting exponent exactly 1/2 if and only if RH, and at least 1/2 unconditionally - reads 0.4413..0.4517 at these depths, which calibrates every reading above: at census mass even the classical meter sits a few hundredths under 1/2.
The distribution of the apparent exponent across designs at fixed base, sorted by the generator: at q = 3 the three columns read 0.4802, 0.4869, 0.4962; at q = 4 the ten run 0.4527 to 0.5197; at q = 5 the twenty-four with nonzero meter run 0.4471 to 0.5358; at base 10 the ten Kempner columns run 0.4465 to 0.5273. All 47 readings sit within 0.054 of 1/2, against cut readings (theta at the checkpoint alone) that scatter over 0.22..0.53 for the same data - the single-cut estimator is noise, the running maximum is the meter.
Digit strings across divisors
The meter weighs mu along a design; this section weighs the design itself against a divisor, the arithmetic input any multiplicative estimate over S_F has to have. Write N_F(L; d, r) for the number of length-L digit strings over F whose value sum_j f_j q^j is r mod d, and N_F(L; d) = N_F(L; d, 0). The value map is injective on strings of one length, so with 0 in F this counts the multiples of d below q^L whose padded digits lie in F, and with 0 outside F it is the length-L block of S_F, the blocks l <= L partitioning S_F below q^L. Write k = |F|, e(x) = exp(2 pi i x), g_F(t) = sum_{f in F} e(f t), Delta_F for the gcd of the digit differences, gamma_F(d) = max over a not 0 mod d of |g_F(a/d)|/k, and normalized error for d |N_F(L; d) - k^L/d| / k^L. Residue distribution of digit-restricted sets is the subject of Erdos, Mauduit and Sarkozy 1998; what follows is derived here from the transform, each statement carrying its own hypotheses. Every number is printed by lab/rho-decoupling.
- Orthogonality. Proved.
N_F(L; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < L} g_F(a q^j/d): expand the divisibility indicator in additive characters modd; the digits are independent, so the character sum factors over positions. Thea = 0term isk^L/dand every bound below is a bound on the rest. - The uniform geometric bound. Proved. For
k >= 2,d >= 2,(d, q) = 1andgcd(d, Delta_F) = 1, everyrand everyL >= 1:|N_F(L; d, r) - k^L/d| <= ((d-1)/d) k^L (1 - 8/(k^2 d^2))^L <= k^L exp(-8 L/(k^2 d^2)). Coprimality toqkeepsa q^jnonzero moddat every position,|g_F(a/d)|^2 = k^2 - 4 sum_{f < f'} sin^2(pi a (f' - f)/d), and ifddivideda (f' - f)for every pair thend/gcd(a, d)would divideDelta_Fand forced | a, so one pair sits at distance>= 1/dfrom an integer andgamma_F(d)^2 <= 1 - 16/(k^2 d^2). The census asserts the weaker form as an exact integer inequality at every cell where the hypotheses hold; the largest observed-to-bound ratio is0.187, atq = 100,F = {0,1},L = 16. The exponentd^(-2)is not slack: atd | q - 1withFan arithmetic progression of common differencem', takinga m' = 1 mod dgives|g_F(a/d)|/k = sin(pi k/d)/(k sin(pi/d)) = 1 - Theta(k^2/d^2). - The dense-digit bound. Proved. For
F = {0..q-1}minusEwithm = |E|,k = q - mand(d, q) = 1:gamma_F(d) <= (d/2 + m)/k, sinceg_Fis the full Dirichlet kernel lessg_E,|D_q(a/d)| <= 1/(2||a/d||) <= d/2and|g_E| <= m; hence ford/2 + m < kthe error is at mostk^L ((d/2 + m)/k)^L, uniform inr. - A power saving at level
q^(1-eps). Proved. Fixeps in (0,1)and takeq >= 4^(1/eps),m <= q^(1-eps)/2,L >= 4/eps. Every2 <= d <= q^(1-eps)coprime toqthen has per-digit factor(d/2 + m)/k <= q^(-eps/2), sosum over those d of |N_F(L; d) - k^L/d| <= k^L q^(1 - eps L/2) <= k^L x^(-eps/4)atx = q^L: a power saving over the whole block, not one divisor at a time. The saving is carried by the digit count and not by the base. At the fixed divisord = 7the per-digit error rate reads0.4869, 0.3312, 0.2484, 0.1104, 0.0167alongq = 3, 4, 5, 10, 100askruns2, 3, 4, 9, 99, against per-factor ceilings0.9010, 0.7490, 0.5617, 0.2002, 0.0221; forF = {0,1}atq = 100it stays0.4992with ceiling0.9010, the same ceilingF = {0,1}has atq = 3. - The split across the base's own divisors. Proved. For
d = d1 d2withd1 | q^mfor somem <= Land(d2, q) = 1, the lowmdigits fix the value modd1and reach the rest only through the invertible multiplierq^m mod d2, soN_F(L; d) = sum over w in F^m with d1 | val(w) of N_F(L - m; d2, r_w),r_w = -val(w) (q^m)^(-1) mod d2. The density splits exactly,rho_F(d1 d2) = (N_F(m; d1)/k^m) (1/d2), and the base part is a digit-string count rather than1/d1: a divisor sharing a factor withqis read off the digits, never off a density. Checked against direct enumeration atq = 6,d = 10. - The digit-gcd hypothesis is a wall, not a convenience. Proved. If
gcd(d, Delta_F) > 1there is no equidistribution at all: atq = 3,F = {0,2},d = 2every value is even,N_F(L; 2) = k^L, and the normalized error is exactly1at everyL. Overd <= 200the unrestricted worst error for that family reads1.0483atL = 32, pinned atd = 164, against0.019166oncedis required coprime toDelta_F. Such families reduce to a primitive one throughS_(aF') = a S_(F'), the scaling map of the transfer above. - The slow column at fixed digit count. Verified. The bound decays in
Lonly, at a rate the digit count controls, and the census sees nothing better:F = {0,1}atq = 100has worst normalized error28.593, 14.590, 9.0340, 7.2034atL = 16, 32, 64, 96overd <= 500, per-digit factor0.9929, with argmaxd = 481 | q^3 - 1at the first two depths andd = 303 | q^2 - 1at the last. Sparse digit sets are outside the reach of every per-divisor estimate here, exactly as they are outside the reach of the exponent census above. - The worst divisor is pinned. Verified. The obstruction is small multiplicative order: the orbit
a q^j mod dvisits onlyord_d(q)points, so no averaging happens across positions, and at every family's deepest level the sweep argmax hasord_d(q) <= 8, hence dividesq^t - 1witht <= 8(d = 164atq = 3,d = 143atq = 10,d = 101, 303, 481atq = 100); shallow depths can stray,d = 199withord = 99atq = 10,L = 6. - The signed pinned sum does not cancel. Verified. Weight each squarefree pinned modulus
e = (q^t - 1)/g,g | q - 1,e >= 2,t <= L, bymu(e), withT_L(e) = N_F(L; e) - k^L/e, and set the signed sumSigma_L = sum mu(e) T_L(e)against the absolute sumAbs_L = sum |T_L(e)|over the same moduli. Printed at everyL = 3..40by the generator, the ratioSigma_L/Abs_Lswings across[-1, 1](-1.00atL = 5, 6in the first family) with no decay:-0.211, -0.123, +0.069, -0.498atL = 10, 20, 30, 40forF = {0,1}atq = 3and+0.812, -0.495, -0.127, -0.192for one excluded digit atq = 10, whileAbs_L/k^Lreads2.1 * 10^-4and3.9 * 10^-12atL = 40; the sum rests on6of29terms in the first family and4of60in the second, the four largest att = 7, 9and att = 5, 7, 8, 10. EveryN_F(L; e)is an exact integer of the carry count, which sums the digits in each residue class of positions modtand counts the targetsj eby carries, polynomial inLat everyt;mu(e)is read off a complete factorisation of everyq^t - 1tot = 40, cyclotomic factors first, then Pollard-Brent, every prime certified by deterministic Miller-Rabin below3.317 * 10^24. The sign ofmuacross the family does not organise the errors: a signed Type I sum over these moduli buys only a bounded factor over the absolute one,|Sigma_L|/Abs_Lreading0.498and0.192atL = 40, and that factor does not grow with depth at any depth computed. - Conjecture. For
F = {0..q-1}minus one digit andd = q^t - 1, the worst orbit-mean damping isk^(-1/t) (1 + o(1)): the orbit carriest - 1undamped points and one damped by~ 1/k. Atq = 100andL = 12the single-divisor probes read orbit mean0.1059atd = q^2 - 1againstk^(-1/2)and0.2369atd = q^3 - 1againstk^(-1/3), with the proper divisord = 3367 | q^3 - 1better at0.0549andd = 101 | q + 1pinned but harmless at0.0261. Two values ofton one base are a check, not a law. - The second moment across residues. Proved.
sum_{r mod d} (N_F(L; d, r) - k^L/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < L} |g_F(a q^j/d)|^2, by Parseval moddon the orthogonality identity: the mean is thea = 0term, the variance is the rest, no cross terms survive. It gives up the supremum overrand buys an average overa, which is the one place a saving can survive at a pinned divisor, where every per-factor bound is flat.
A power saving under GRH at large base
The census above measures cancellation and proves none of it. This section proves some, at the opposite end of the digit scale: not the sparse columns of the census but the dense ones, the base taken large and a single digit removed. There the mass exponent alpha_q = log_q k sits just under 1, the indicator of S_F opens into additive frequencies by the same orthogonality the divisor section uses, and each frequency carries a Mobius exponential sum, which under the generalized Riemann hypothesis is x^(3/4 + eps) uniformly in the frequency. The entire cost of the expansion is one l^1 norm, and past a computable base that cost is smaller than the mass. What comes out is a bound of Mertens shape read against the set's own counting function, conditional and dense-only, with both of those limits proved rather than assumed. Every constant, table row, margin and rung below is printed or test-pinned by lab/mertens-numerology.
Proved under GRH below is the Proved tag with the hypothesis written inside the statement: steps 1, 2, 3 and 5 are derived here from definitions, and step 4 is one published theorem, quoted at its source and used exactly as stated.
- The setting.
Eis the excluded digit set withm = |E| >= 1,F = {0..q-1}minusE,k = q - mandalpha_q = log k / log q, soA_F(x) >>_q x^(alpha_q)at everyxby the counting identities of the first section. The digit symbol isg_F(t) = sum_{d in F} e(d t)of the divisor section andD_q(t) = sum_{d = 0}^{q-1} e(d t)is the full Dirichlet kernel,|D_q(t)| = |sin(pi q t)/sin(pi t)|. WriteD_Lfor the length-Ldigit strings overFandhat F_L(t) = sum_{n in D_L} e(n t)for the level-Ltransform, which factors asprod_{j < L} g_F(q^j t)because the digits are independent. The one-step constant of the shifted-grid recursion isB_q(F) = sup_t sum_{r mod q} |g_F((t+r)/q)|, bounded above byq PB_q(m)in step 3, with the proved constantPB_q(m) = sqrt(m) + Phi_q/q, wherePhi_q = (4/pi) q + (2q/pi) H(ceil((q-2)/2)) + (1 - 2/pi)(q - 2) + 0.727andH(n) = ln n + gamma + 1/(2n); the exponent cost isc_q = log PB_q(m)/log q, defined from that proved bound and never from the exact supremum. - Theorem. Proved under GRH. Assume the generalized Riemann hypothesis in its Dirichlet form:
L(s, chi)has no zero in the half planesigma > 1/2, for every Dirichlet characterchiof every modulus. Letq >= 3690, letFomit exactly one digit, and leteps > 0. Then|M_F(x)| <<_{q,eps} x^(3/4 + c_q + eps)for allx >= 2; and3/4 + c_q < alpha_qat every suchq, so withdelta_q = (alpha_q - 3/4 - c_q)/alpha_q > 0the same bound reads|M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q + eps), a power saving against the set's own mass. The implied constant depends onqand onepsand on nothing else; no uniformity inqis claimed anywhere.delta_qis fixed beforeepsis chosen, so the statement delivers every fixeddelta' < delta_qand never the endpointA_F(x)^(1 - delta_q). - Step 1, orthogonality. Proved. For
0 <= n < q^L,1_{D_L}(n) = q^(-L) sum_{0 <= a < q^L} hat F_L(a/q^L) e(-n a/q^L)by completeness of the additive characters modq^L, andhat F_Lfactors over digit positions: the identity of the divisor section with the modulusq^Lin place ofd, read as an expansion rather than as a count. - Step 2, the
l^1recursion. Proved. Putc_L = sum_{0 <= a < q^L} |hat F_L(a/q^L)|and splita = a' + s q^(L-1). The transform peels at the positionj = 0,hat F_L(t) = g_F(t) hat F_{L-1}(q t), sosmoves that factor alone andhat F_{L-1}is1-periodic; the inner sum overs mod qis a shifted grid ofqpoints, andc_L = sum_{a'} |hat F_{L-1}(a'/q^(L-1))| sum_{s mod q} |g_F((a'/q^(L-1) + s)/q)| <= B_q(F) c_{L-1}. Hencec_L <= B_q(F)^Land the normalizedl^1mass isq^(-L) c_L <= (B_q(F)/q)^L: one constant per digit, no interaction between positions. - Step 3, the kernel bound. Proved.
|g_F| <= |D_q| + |g_E|splitsB_q(F)into a kernel part and an excluded part. Theqpoints(t + r)/qare spaced1/q; writingd_rfor the distance of each toZ,|D_q((t+r)/q)| = |sin(pi q d_r)|/sin(pi d_r), the two points nearest the singularity contribute at most(4/pi) q + 0.727by the two elementary inequalitiessin(pi v) <= 4v(1-v)on[0, 1/2]and1/sin x <= 1/x + 1 - 2/pion(0, pi/2](the first because4x(1-x) - sin(pi x)splits into a concave and a convex piece with the right signs, the second because1/sin x - 1/xincreases), and the remainingq - 2points pair off at distances>= j/qand contribute at most(2q/pi) H(ceil((q-2)/2)) + (1 - 2/pi)(q - 2). Sosup_t sum_{r mod q} |D_q((t+r)/q)| <= Phi_q. The excluded part is exact rather than estimated: the excluded digits are distinct modq, so Parseval on the shifted grid givessum_{r mod q} |g_E((t+r)/q)|^2 = q mfor everyt, and Cauchy-Schwarz turns that intosum_{r mod q} |g_E((t+r)/q)| <= q sqrt(m). HenceB_q(F) <= q PB_q(m)and the normalizedl^1mass of step 2 is at mostq^(L c_q). - Step 4, the Mobius input. Quoted. Under GRH,
max_{theta in [0,1)} |sum_{n <= x} mu(n) e(n theta)| <<_eps x^(3/4 + eps). This is the casea = 1/2of Baker and Harman 1991, whose hypothesis is exactly thatL(s, chi)is zero-free insigma > afor every Dirichlet character, whose implied constant depends only oneps, and whose maximum is over all realtheta; the frequencies this proof uses are thea/q^l, well inside that uniformity. The statement is restated at source in Porritt 2018 and in Zhang 2024. This is the one step not derived here. - Step 5, assembly and the wall. Proved. If
0 in F, thenS_FbelowxisD_Lless{0}intersected with[1, x]atL = ceil(log_q(x+1)), and steps 1 to 4 apply once:|M_F(x)| <= (B_q(F)/q)^L max_theta |sum_{n <= x} mu(n) e(n theta)| <<_{q,eps} x^(3/4 + c_q + eps). If0is not inF, thenS_Fbelowq^Lis the disjoint union of the exact-length blocksl <= L, and summing the per-block bounds is a geometric sum of ratioq^(3/4 + c_q + eps) > 1, so the top block sets the exponent and the answer is the same; the statement holds at allx >= 2because the implied constant absorbs the bounded range whereLis small. Converting to theA_Fyardstick needs3/4 + c_q < alpha_q, equivalently the constant-space certificategap_q(m) = (q - m) q^(-3/4) - PB_q(m) > 0. Atm = 1that certificate is negative at every3 <= q < 3690and positive atq = 3690(Verified, exhaustive in the generator), and steps up at everyq >= 723by the monotone floor below, soq >= 3690is a half line and not a window. - What the proof does not use. Proved. No zero-density input, no restriction of
xto a power ofq, no multiplicative structure ofS_F(there is none: the first section's4 x 13witness), and nol^1bound quoted from the literature. Step 3 is self-contained and explicit at every base, which is what the theorem needs and what the sharper base-10l^1bound behind Maynard 2019 does not offer at generalq. - Corollary,
mexcluded digits. Proved under GRH. With|E| = mand the same constants, the five steps run unchanged wheneverPB_q(m) < (q - m) q^(-3/4), and give|M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q + eps)withdelta_q = (alpha_q - 3/4 - c_q)/alpha_q. Under the proved constants that condition holds form <= 6atq = 10^4,m <= 78atq = 10^5andm <= 451atq = 10^6(Verified, each maximum asserted maximal in the generator), againstsqrt(q) = 100, 316, 1000, and it holds asymptotically form <= q^(1/2)(1 - o(1))sincePB_q(m)issqrt(m)plus a term of size(2/pi) ln q. The squarefree-digit-gcd hypothesis carried by the exponent conjecture below is automatic in this regime and is not dropped:m < floor(q/2)leaves two consecutive digits inF, sogcd(F) = 1and the vanishing family of the transfer section cannot occur; at smallkthe hypothesis must be stated. - The shape at large base. Proved under GRH.
Phi_qis(2/pi) q ln qup to lower order, soc_q = (ln ln q + ln(2/pi) + o(1))/ln q -> 0whilealpha_q -> 1, hencedelta_q -> 1/4and|M_F(x)| <<_{q,eps} A_F(x)^(3/4 + o(1)): the full-line GRH exponent transplanted verbatim onto the digit-restricted column, measured against that column's own mass. The convergence is logarithmic and nothing better;c_qtracks(ln ln q + ln(2/pi))/ln qto within0.01atq = 10^12(Verified, the generator). - The constants. Verified. The generator prints
alpha_qtruncated down at six digits,c_qrounded up at five anddelta_qrounded down at five, each from the unrounded value with a directional guard of10^-12, so every printed digit is a true bound in its own direction andalpha_qnever prints as1.000000; the scientific rows carry a relative guard of10^-10. Both forms of the test,3/4 + c_q < alpha_qandgap_q(m) > 0, are computed and their agreement asserted at every row and across3 <= q < 20000.
q | alpha_q | c_q (proved, up) | delta_q (down) | closes |
|---|---|---|---|---|
| 1000 | 0.999855 | 0.28087 | -0.03102 | no |
| 2000 | 0.999934 | 0.26335 | -0.01342 | no |
| 3000 | 0.999958 | 0.25430 | -0.00434 | no |
| 3689 | 0.999966 | 0.24997 | -0.00001 | no |
| 3690 | 0.999967 | 0.24997 | 0.00000 | yes |
| 5000 | 0.999976 | 0.24393 | 0.00605 | yes |
| 10^4 | 0.999989 | 0.23141 | 0.01858 | yes |
| 10^5 | 0.999999 | 0.19906 | 0.05094 | yes |
| 10^6 | 0.999999 | 0.17589 | 0.07411 | yes |
| 10^9 | 0.999999 | 0.13305 | 0.11695 | yes |
- The margin at the wall. Verified. The saving at
q = 3690is far below the fifth printed digit, so the rounded columns cannot display its sign and never certify it. The certificate is the pair of scientific bounds printed from the cancellation-reduced formdelta_q = ln(1 + gap_q(m)/PB_q(m))/(alpha_q ln q), which never differences two numbers of size1to reach one of size10^-6:delta_q <= -2.395807653 * 10^-6andgap_q(1) <= -1.533059397 * 10^-4atq = 3689, againstdelta_q >= 5.863425182 * 10^-6andgap_q(1) >= 3.752213034 * 10^-4atq = 3690. Beyond the wall the gap rises at every one of the96310steps of3690..10^5, the smallest step being>= 0.00003172at the top of that range, where theq^(-3/4)growth of the mass term is nearest the4/(pi q)jump of the harmonic term. - The ladder. Proved under
Z(a). WriteZ(a), for1/2 <= a < 1, for the hypothesis thatL(s, chi)has no zero insigma > afor every Dirichlet character;Z(1/2)is GRH. AssumeZ(a), letFomit exactly one digit and letq >= q_0(a), the least base withPB_q(1) < (q-1) q^(-b(a)). Then for everyeps > 0and allx >= 2,|M_F(x)| <<_{q,eps} x^(b(a) + c_q + eps)andb(a) + c_q < alpha_q, so|M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q(a) + eps)withdelta_q(a) = (alpha_q - b(a) - c_q)/alpha_q > 0; the corollary runs atmexcluded digits wheneverPB_q(m) < (q-m) q^(-b(a)). The proof is the one above with a single substitution: step 4 quotes the exponentb(a)thatZ(a)buys, and steps 1, 2, 3 and 5 never name an exponent, the geometric sum of step 5 still having ratio above1. Sincealpha_q -> 1andc_q -> 0whileb(a) < 1is fixed, every common zero-free half plane for Dirichlet L-functions buys a power saving over the dense column, and GRH is only its first rung: the price of a weaker hypothesis is paid entirely in the base. - The input
b(a), and where it comes from. Proved.b(a)is the smaller of two quoted tables: Baker and Harman 1991 givesa + 1/4on1/2 <= a < 11/20,4/5on11/20 <= a < 3/5and(a+1)/2on3/5 <= a < 1, and Zhang 2024, Theorem 1.1, gives(8a - 7a^2)/(4 - 2a)on1/2 <= a <= 4/7. Where both apply Zhang is smaller and the two meet exactly at the ends of the overlap, by two factorisations:Zhang(a) - (a + 1/4) = -5(a - 1/2)(a - 2/5)/(4 - 2a)is negative on(1/2, 11/20)andZhang(a) - 4/5 = -7(a - 4/7)(a - 4/5)/(4 - 2a)is negative on[11/20, 4/7), with equality ata = 1/2(both3/4) and ata = 4/7(both4/5); andb(a) >= 3/4on the whole range, Baker-Harman by inspection and Zhang byZhang(a) - 3/4 = -7(a - 1/2)(a - 6/7)/(4 - 2a) > 0on(1/2, 4/7]. Verified in the generator over every rational of denominator<= 200inside the overlap, in exact integer arithmetic,b(a)carried as a rational and compared by cross multiplication throughout. - The rungs. Verified. Each row names its
aand the table the exponent comes from;bothmeans the two tables agree there, and a rung is meaningless quoted without them.q_0(a)is the leastq >= 3withgap_q(a, 1) = (q-1) q^(-b(a)) - PB_q(1) > 0andQ(b)the proved monotone floor below. A wall prints as an exact integer only when it sits below2^53and both neighbouring gaps exceed1024ulps of the terms differenced; otherwise the row prints<=and a scientific upper bound, which is a bound on the leastqand not the leastq. Every wall below4 * 10^6is reproduced by an exhaustive scan fromq = 3against the bisection.
a | b(a) | source | q_0(a) | Q(b) |
|---|---|---|---|---|
| 1/2 | 3/4 | both | 3690 | 723 |
| 13/25 | 1417/1850 | Zhang | 8578 | 1486 |
| 11/20 | 913/1160 | Zhang | 33547 | 4754 |
| 4/7 | 4/5 | both | 92317 | 11221 |
| 3/5 | 4/5 | BH | 92317 | 11221 |
| 2/3 | 5/6 | BH | 3107080 | 216023 |
| 3/4 | 7/8 | BH | 6939524168 | 129458304 |
| 4/5 | 9/10 | BH | <= 3.09358e13 | 128606353005 |
| 9/10 | 19/20 | BH | <= 3.23663e34 | <= 1.73431e28 |
| 19/20 | 39/40 | BH | <= 9.24614e83 | <= 3.30712e68 |
- The floor is proved, not scanned. Proved. Per step
PB_{q+1}(1) - PB_q(1) < 1.291/(q-2)forq >= 40: the harmonic term jumps by at most(4/pi)/(q-2), the(1 - 2/pi)(q-2)/qterm adds under0.017/(q-2)and the0.727/qterm falls, and a step that does not jump the harmonic term is net negative. The mass term(q-1) q^(-b)gains at least(1-b)(q+1)^(-b)per step, sogap_q(a, 1)steps up wherever(1-b)(q-2)(q+1)^(-b) >= 1.291, a quantity strictly increasing inq;Q(b)is the leastq >= 40where it holds, and the gap steps up at everyq >= Q(b). Below it nothing closes:gap_q(a, 1) < 0on3 <= q < 3690at every rung (exhaustive), and on[3690, Q(b)]the smooth majorantU(q) = q^(1-b) - PB_q^-(1)dominates the gap and has exactly one interior minimum, sinceU'(q) = (1-b) q^(-b) - (2/pi)/(q-2) - 2(1 - 2/pi)/q^2is positive exactly when a quotient falling strictly from+infto0drops below1, so its maximum on any interval sits at an endpoint and both endpoints are negative.Q(b) < q_0(a)at every rung, so each printed wall is the leastqand the gap steps up from it on, with no sweep needed at any rung. - The floor and the wall hold at every
bin[3/4, 1), not only at the printed rungs. Proved. Below3690the one exhaustive scan covers everybat once:q^(-b)falls inb, sogap_q(b, 1) <= gap_q(3/4, 1) < 0on3 <= q < 3690, that range being cleared exhaustively atb = 3/4. The floor itself rises withb, since(1-b)(q-2)(q+1)^(-b)falls inbat fixedq, soQ(b) >= Q(3/4) = 723. At the floor, minimality ofQ = Q(b)bounds(Q-1) Q^(-b) < 1.291/u + 0.015above, the slack2 Q^(-b) < 0.015coming fromQ >= 723, andln(Q+1) > ln(1.291/u)/ubelow, both in terms ofu = 1 - balone; feeding them into the lower boundPB_Q^-(1)throughln((Q-2)/2) >= ln(Q+1) - ln(1448/721)and(Q-2)/Q >= 721/723givesgap_Q(b, 1) < [1.291 - (2/pi) ln(1.291/u) - 2.544 u]/u, the coefficient2.544assembled from those three ingredients,0.015,ln(1448/721)and721/723. Its bracket increases on(0, 1/4]and so is at most its value-0.39014atu = 1/4, hencegap_Q(b, 1) < -1.56. On[3690, Q]the majorant differs from the gap by under0.004, soU(Q) < -1.556, whileU(3690, b)falls inbwithU(3690, 1417/1850) < -0.95, and anybbelow1417/1850hasQ(b) <= 1486 < 3690and an empty range. Sogap_q(b, 1) < 0on[3, Q(b)]and steps up fromQ(b)on at everyb:q_0(a)exists and exceedsQ(b)at everya, printed rung or not. Constants Verified in the generator on theb-grid0.75..0.975in steps of0.005. - What a weaker half plane spends first. Verified. The
m-budget atq = 10^7is the largestmwithPB_q(m) < (q-m) q^(-b(a)), printed by the generator for the rungs whose wall lies below10^7. Each row carries itsaand its source, a rung quoted bybalone being meaningless: two rungs shareb = 4/5from different tables and the budget, not the theorem, is what a wider zero-free half plane costs.
a | b(a) | source | max m |
|---|---|---|---|
| 1/2 | 3/4 | both | 1971 |
| 13/25 | 1417/1850 | Zhang | 1002 |
| 11/20 | 913/1160 | Zhang | 365 |
| 4/7 | 4/5 | both | 176 |
| 3/5 | 4/5 | BH | 176 |
| 2/3 | 5/6 | BH | 8 |
- The cost-out against a hypothetical Type I defect. Verified. A conditional Type I argument over
S_Fwould run against a level-x^(alpha_q/2)distribution bound for the digit strings whose error carries a defectx^(m/(2(q-m) ln q)), a bound this page does not state, and would have to pay that defect out of the saving proved here, so the generator sets the two exponents side by side. They sit on different yardsticks and no derivation joins them:delta_qis normalised to the mass, so as a power ofxthe saving isx^(alpha_q delta_q)withalpha_q >= 0.99993on every row compared, while the defect multipliesk^L. On those rows the saving is below the defect at the wall (5.86342 * 10^-6against1.65022 * 10^-5atq = 3690, a factor above2.8) and above it fromq = 3692on, the least such base in a scan of3690..10^5where the difference rises at all96310steps, monotonicity beyond the scan not being proved; byq = 10^9the saving1.16951 * 10^-1clears the defect2.41275 * 10^-11by over nine decades, and the tightest corollary row,q = 10^6atm = 451, clears its own defect1.63296 * 10^-5at3.14081 * 10^-5. The whole failure at the wall is the two steps3690, 3691, so a sharper constant that moves the wall moves the comparison too and must be re-costed rather than inherited. - Conjecture. That such a defect is absorbed at all. The comparison above is two exponents from two unrelated statements on two yardsticks; it is not a necessary condition, no theorem about
M_Ffollows from it, and the string-to-interval bookkeeping and the bilinear half of any such argument are untouched here. - The
l^1floor, and what it forecloses. Proved. For every digit set,sum_{r mod q} |g_F((t+r)/q)|^2 = q kexactly, sosum_{r mod q} |g_F((t+r)/q)| >= q k / max_r |g_F| >= qfor everyt: the recursion of step 2 never contracts,B_q(F) >= q, andc_q >= 0at every base and every digit set, so a negativec_qis an arithmetic error and not a discovery. The consequence is a hard limit on this decomposition, not on the problem: it needsalpha_q > 3/4, that isk > q^(3/4), so every column at fixed digit count is out of its reach,F = {0,1}at base 3 included, under GRH or without it. The dense columns this section proves something about and the sparse columns the census measures do not overlap. - Nothing unconditional follows in this decomposition. Proved. Put Davenport's unconditional
max_theta |sum_{n <= x} mu(n) e(n theta)| <<_A x (log x)^(-A), carried at source in Porritt 2018, into step 4: by thel^1floor the result is at best of sizex (log x)^(-A), which exceedsA_F(x)by the powerx^(1 - alpha_q). An unconditional power saving here would need an unconditional uniform power-saving input, which is itself of zero-free-strip strength; an unconditional route has to split arcs and use the structure ofmuin progressions modq^j, which this decomposition never touches. - The
l^2route is worse than trivial. Proved. Cauchy-Schwarz with Parseval on both factors,sum_{a mod q^L} |hat F_L(a/q^L)|^2 = q^L k^Landsum_{a mod q^L} |sum_{n <= q^L} mu(n) e(n a/q^L)|^2of size(6/pi^2) q^(2L), gives exponent(1 + alpha_q)/2 > alpha_q. The supremum over frequencies paid against thel^1mass is the only arrangement of this decomposition that saves anything. - The ceiling, and the endpoint. Proved. Even with the conjectured
x^(1/2 + eps)in step 4, the exponent Porritt 2018 records as the expected one, the floorc_q >= 0still forcesalpha_q > 1/2, that isk > q^(1/2): the exponent conjecture below, which is about fixedk, is beyond every version of this method and not merely beyond its conditional form. And within the dense regime the endpoint stays out:delta_qis fixed beforeeps, so what is proved isA_F(x)^(1 - delta')for every fixeddelta' < delta_qand neverA_F(x)^(1 - delta_q), a distinction that is the whole claim atq = 3690, wheredelta_q >= 5.863425182 * 10^-6. - What this is, against the literature. Verified. As far as the sources in REFS.md are read, none of them carries a Mobius or Mertens sum over a digit-restricted set: the nearest multiplicative function computed over a missing-digit set is the divisor function (Kim 2024), whose own framing is that the set's lack of multiplicative structure blocks the standard approaches, and the nearest arithmetic-function theorem over such a set is the prime count of Maynard 2019, which enters through the set's level of distribution and not through a Mobius bound. The theorem above is of Mertens shape: a power of the set's own counting function,
A_F(x)^(1 - delta')for every fixeddelta' < delta_q, at everyx >= 2, for one excluded digit at everyq >= 3690and formexcluded digits under the stated condition, the bound it beats being the trivial|M_F(x)| <= A_F(x)on those columns. The card carries both halves: it is conditional on GRH, it yields nothing unconditional inside this decomposition, and it says nothing whatever in the sparse regimek <= q^(3/4)where the census and the exponent conjecture live. - The lane. This section is written up on the shelf as sparse-mertens-under-grh: the theorem, the corollary, the ladder and the
l^1floor with full proofs, and itsscripts/verify.pyrecomputes every constant, table row, margin and rung above from the formulas alone, independently oflab/mertens-numerology, in under three seconds.
The pair route
The section above buys a power saving on the dense columns under GRH and states plainly that nothing unconditional follows from that decomposition: an unconditional route has to split arcs and use the structure of mu in progressions mod q^j. This section is that route, laid out as far as it goes. It follows the only existing proof that counts a thin arithmetic sequence on a missing-digit set, Maynard 2019, and asks what changes when the sequence counted is mu rather than the primes. Most of that chain never looks at the sequence at all; the two steps that do are proved here; what is left is a threshold on one number the digit set owns. No theorem about M_F comes out of it. A criterion does, and the criterion fails at base 10 on one number, the exceptional-set threshold, and holds from base 21.
Notation as in the divisor section, with x = q^L, D_L the length-L digit strings over F, hat F_L(t) = sum_{n in D_L} e(n t) = prod_{j < L} g_F(q^j t), alpha = log_q k the mass exponent, S_mu(t) = sum_{n <= x} mu(n) e(n t), rad(q) the product of the primes dividing q and omega(n) the number of them. Write alpha_1 for the l^1 exponent of the transform in its sup-over-shift form, sup_beta sum_{a < Y} |hat F_l(beta + a/Y)| << k^l Y^(alpha_1), which is 27/77 at base 10 with one digit excluded. That is the strength the source's own l^1 lemma carries and the strength Farey spacing consumes; it dominates the bare grid exponent, so every lower bound on the grid exponent below transfers up to it, and the threshold it is asked to clear is correspondingly the stronger ask. Call e base-smooth when rad(e) divides rad(q), and write (E1) for the hypothesis that every prime dividing the gcd of the digit differences of F divides q: the one-dimensional form of condition (E) of coprime, and the hypothesis Lemma A' there consumes.
- The bilinear half of the chain never sees the coefficients. Verified. The Type II estimate of that proof is stated for arbitrary
1-bounded sequences with one support constraint, that every counted integer carries a divisor in a prescribed dyadic range; its proof applies Cauchy-Schwarz in the long variable first and then drops all four coefficient factors by the triangle inequality, leaving a sum over pairs of frequencies with no coefficient in it at all, which a geometry-of-numbers argument places near a rank-2 lattice or on a line. Residue sums of the coefficient side occur exactly once in that proof, on the major arcs at moduli below a fixed power oflog x. So a Type II estimate on a digit set is not a hypothesis about cancellation of the coefficients in progressions, and the whole range above that cut transfers from primes tomuunread, every sentence of this bullet read at its source. - The
l^1floor is the shifted-grid floor iterated. Proved. Thel^1floor of the GRH section,sum_{r mod q} |g_F((t+r)/q)| >= q k / max_r |g_F| >= qfor everyt, is one digit position of the same statement; iterating it overLpositions through the peeling recursion of that section's step 2, or reading it off the grid directly bysum_a |z_a| >= (sum_a |z_a|^2)/max_a |z_a|with Parsevalsum_{a mod q^l} |hat F_l(a/q^l)|^2 = q^l k^land the maximumk^lata = 0, givessum_{a mod q^l} |hat F_l(a/q^l)| >= q^land hencealpha_1 >= 1 - alphaat every base and every digit set. One mechanism, stated once there per position and once here per exponent. Two consequences: anl^1exponent below1/2forcesk > sqrt(q), so the sparse columns of the census are outside this route exactly as they are outside the route of the GRH section; andalpha + alpha_1 >= 1always, which is what makes the scale sum in the level-of-distribution statement below geometric with ratio at least1. - No exceptional character sits at a base-smooth modulus. Proved. Every real primitive Dirichlet character of base-smooth modulus has conductor dividing
8 rad(q), and the conductors in play number exactly2^omega(q_1)at oddqand3 * 2^omega(q_1)at evenq,q_1the odd part ofq, while the characters number2^omega(q_1)at oddqand4 * 2^omega(q_1)at evenq. A real primitive character of conductorf > 1is the Kronecker symbol of a fundamental discriminant of absolute valuef, so writingf = 2^u f_1withf_1odd,f_1is squarefree anduis0,2or3; base-smoothness forcesf_1 | rad(q), hencef | 8 rad(q). Conversely every2^u f_1of that shape occurs, and the two counts differ: exactly one of+-f_1is1 mod 4, giving one character atu = 0; exactly one of+-f_1is3 mod 4, giving one atu = 2; and both of+-2 f_1are2 mod 4and squarefree, giving two atu = 3, so four characters sit over three conductors for each odd squarefreef_1dividingrad(q), while at oddqonlyu = 0is available and the counts coincide. That is the whole content of the remark in the source that its major-arc moduli are too composite for Siegel zeros to matter: an exceptional zero belongs to a real primitive character, a real primitive character has a fundamental discriminant for a conductor, a fundamental discriminant is squarefree away from a factor4or8, and a power of the base is as far from squarefree as an integer gets. The conductors in play run over a set of size bounded in terms ofqalone rather than to infinity, so Siegel's theorem is never invoked and the constants below are effective. - The major arcs for
mu, with the exponent they deliver. Proved. Letq >= 3, letFsatisfy (E1), letC > 0, and putT = (log x)^CandM(C) = {a mod x : |a/x - b/d| <= T/x for some d <= T and some b coprime to d}. Then there arec > 0andx_0, both depending only onq,kandCand both effectively computable, with|x^(-1) sum_{a in M(C)} hat F_L(a/x) S_mu(-a/x)| <= k^L exp(-c sqrt(log x))forx >= x_0. The proof splitsM(C)at the base-smooth denominators. Off them the modulus carries a factord_2 > 1coprime toq, and the perturbed Lemma A' of coprime gives|hat F_L(a/x)| <= k^L exp(-c' log x / log log x)against the trivial|S_mu| <= x. On themx = q^Lmakes every suchb/dan exact grid point, so the arcs are intervals of consecutive integers and no Dirichlet approximation enters; there|hat F_L| <= k^Lis trivial, partial summation strips the shift, and what is left issum_{n <= u, n = r mod e} mu(n)at a base-smoothe <= T, which the classical zero-free region forL(s, chi)bounds byu exp(-c'' sqrt(log u))(Davenport, chapters 14 and 20) with the only ineffective ingredient, the exceptional real zero, removed by the conductor bound above and the effective Landau-Page bound of the same chapter 14. There is no main term at any arc, the frequencya = 0included, where the contribution isk^L M(x)/x. The saving isexp(-c sqrt(log x)). It is not compared with the(log x)^(-C)the source states for the prime analogue, which is an asymptotic with a main term where this is a bound with none; what is worth stating is that the main term is absent at every arc and that the prime number theorem is what puts thea = 0term inside the error. - The level of distribution on an initial segment. Proved. Assume (E1) and the large sieve the design supplies,
sum_{d <= Q} sum_{(b,d) = 1} |hat F_m(b/d)| << k^m (Q^(2 alpha_1) + Q^2 q^(-m(1 - alpha_1)))at everym <= L, which follows from thel^1exponent by Farey spacing alone and readsQ^(54/77) + Q^2 Y^(-50/77)at base 10. Then for everyB > 0there isCwithsum_{d <= Q, (d,q) = 1} max_{y <= x} |#{n in D_L : n <= y, d | n, (n,q) = 1} - (1/d) #{n in D_L : n <= y, (n,q) = 1}| <= k^L (log x)^(-B)at everyQ <= x^(1 - alpha_1) (log x)^(-C). The initial segment costs nothing in the level and one power oflog xin the saving, for two reasons.D_Lbelowyis a disjoint union of blocks{P q^m + t : t in D_m}, at mostkof them per scale whateveryis; and the error the transform gives for#{t in D_m : t = r mod d}is uniform in the target residuer, so a shifted target is exactly as cheap as the residue0the source asks for. Above the cut the large sieve pays, below it Lemma A' pays, and the scale sum is dominated by its top scale becausealpha + alpha_1 >= 1. - What the base's own divisors cost. Proved. For
d = d_1 d_2withd_1base-smooth and(d_2, q) = 1, the split of the divisor section carries the level tod: the low digits fixn mod d_1and reach the rest only through an invertible multiplier, so the count reduces to the same transform estimate ind_2. What does not carry is the main term. It is a digit-string count times1/d_2and not1/d, readingk^(-v)against a naiveq^(-v)atd_1 = q^v, so a Type I sum with coefficientsc_dproducessum_d c_d rho_F(d)where the coprime case producessum_d c_d / d, and nothing here shows the first small. Nor is the coprimality peeled off in general:sum_{n in S_F, n <= x} mu(n) = sum_{w | rad(q)} mu(w) sum_{n' : w n' in S_F, (n', q) = 1} mu(n')is an identity, and whether it reduces the problem depends on the inner sets. Sometimes it does - atq = 10andF = {0,1}the carry-free scaling of the transfer section gives{n : 2 n in S_F} = S_{0,5}and{n : 5 n in S_F} = S_{0,2}, both designs, and that column sits below this route's ownl^1floor in any case,k = 2 < sqrt(10). Sometimes it does not: atq = 10andF = {0,1,2}the set{n : 2 n in S_F}begins1, 5, 6, 10, 11, 50, 51, 55, 56, 60, 61, 100, 101, 105, and it is a digit design at no base tested, the base-10 digit set it forces being{1,5,6}, which misses10, or{0,1,5,6}, which wrongly admits15because30leavesS_F. The hypothesis(n, q) = 1therefore stays inside the criterion below. - From strings to the design. Proved. The two statements above count
D_L, the padded strings, while the criterion countsS_F. With0 in Fthe two agree belowq^Lbut for the element0, which carriesmu(0) = 0. With0outsideF,S_Fbelowq^Lis the disjoint union of the exact-length blocks, each of them aD_l, so both statements sum overlwith the top block setting the exponent, the geometric sum having ratiok > 1; that the level-of-distribution statement holds on an initial segment is what makes the sum legitimate at everyl. - The window and the criterion, as arithmetic. Proved. Write
betafor the exceptional-set threshold,E = {a : |hat F_L(a/x)| >= k^L x^(-beta)}, andm_tfor thel^texponent of the transform. Both places the source spends the exceptional set reduce tom_t < (2 - t) betafor somet in [1, 2), so the least admissible threshold isinf_{1 <= t < 2} m_t/(2 - t), which at the source's ownt = 235/154andm_t = 59/433is9086/31609 = 0.287449, rounding up to its23/80. The Type II window is then[(5/4) beta, 1 - 2 beta], and by the symmetry of the phase in its two variables also[2 beta, 1 - (5/4) beta]; atbeta = 23/80that is[9/25, 17/40]. Decomposingmuby a Heath-Brown identity of order above1/alpha_1, a piece with a free variable abovex^(alpha_1)is Type I at the level above, and otherwise greedy accumulation lands in the window under two conditions,beta <= 1/4, which merges the two windows into one interval, andalpha_1 + (5/2) beta <= 1, the greedy overshoot. The bilinear estimate itself asks five more, listed in the lattice bullet below; every one of them is free underalpha_1 < 1/3,beta <= 1/4and thel^1floor. Takingt = 1, sobeta = alpha_1 + eps, the binding condition isalpha_1 < 1/4. - The hybrid bound the lattice branch needs holds at every base, with the exponent the digit set owns. Proved. The published proof reaches its bilinear estimate through one bound whose whole purpose is to beat the plain
l^1exponent in the modulus aspect: forD, E, Y, Q_1powers ofqwithD E << Y,q_1 ~ Q_1coprime toqandd ~ Dbase-smooth,sum_{q_2 ~ Q_2, (q_2,q) = 1} sum_{a < d q_1 q_2, (a, d q_1 q_2) = 1} sum_{|eta| <= E/Y, (eta + a/(d q_1 q_2)) Y in Z} F_Y(a/(d q_1 q_2) + eta) << (D E)^(alpha_1) (Q_1 Q_2^2)^(1 - alpha) + E^(alpha_1 + alpha/2) D^(1 + alpha/2) Q_1 Q_2^2 Y^(-alpha/2), whereF_Yis the transform normalised by its own mass. The proof is the source's, carried in general parameters: the product identityF_(Y_1 Y_2)(t) = F_(Y_1)(t) F_(Y_2)(Y_1 t)and the monotonicityF_Y <= F_UforU <= Ysplit the sum, the Chinese remainder theorem sends the residues through complete reduced systems exactly once, thel^1exponent and the shifted large sieve it supplies by Farey spacing pay three of the four factors, and the fourth is pure Parseval on a windowR = q^r,int_0^1 F_R^2 = R^(-alpha)andint_0^1 (F'_R)^2 << R^2 R^(-alpha), the first exact when0is inFand≍otherwise, in the direction used either way. So the two exponents are the digit set's own dimension and nothing else: the modulus exponent is1 - alphaand the saving exponent isalpha/2. At base 10 with one digit excluded the source prints them as1/21and10/21on the single check20/21 < log 9 / log 10, and1 - alpha = 0.045757sits under1/21 = 0.047619whilealpha/2 = 0.477121sits over10/21 = 0.476190: both roundings are safe and both are lossy. By thel^1floor above,1 - alpha <= alpha_1at every base and every digit set, so this bound never loses to the plain one in the modulus aspect, and the reading that a general base must fall back onalpha_1there mistakes the floor's equality case for the general value. - The lattice branch transfers to every base, and the conditions it asks are free below
1/3. Proved. With that bound the whole lattice half of the bilinear estimate runs in general parameters: forx = q^Land the windowN K >= x^(1 - 2 beta),delta >= N/x,Q <= x^(1/2), the sum ofF_x(a_1/x) F_x(a_2/x)over pairs whose large contribution comes from a rank-2 lattice is<< (log x)^5 (Q + E)^(-eps/4) x/(N K), the source's own log power, the count of pairs of base-smooth moduli beingO(Q_0^(eps/2))at every fixed base. Five inequalities close it:2 alpha_1 < alpha;(2 - alpha) 2 beta < 1 - alpha_1; someuin(0, min(1, 2 alpha_1/alpha)]has2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2;5 beta < 1 + alpha/2; and2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2). Their provenance is not uniform: the source writes a numerical check for the second and the fourth, while the first, third and fifth are read off steps it performs silently under the phrase that the exponents have been simplified for an upper bound. Its third written check renames an exponent and imposes nothing past the fourth condition, sobeta < alpha/2is not a hypothesis of the branch. All five are monotone, worse asbetaoralpha_1grows and better asalphagrows, so the corneralpha = 1 - alpha_1,beta = 1/4decides them all, and there they readalpha_1 < 1/3,1/3,1/3,1/2and1 - 1/sqrt(3) = 0.422649. Underalpha_1 < 1/3,beta <= 1/4and thel^1flooralpha + alpha_1 >= 1every one of the five holds, and1/3is sharp: three of them are equalities there. The floor and the threshold onbetaalone are not enough, asalpha_1 = 0.40,alpha = 0.60,beta = 1/4shows, where the first three read0.8 < 0.6,0.7 < 0.6and0.4 < 0.3. So the criterion's ownalpha_1 < 1/4clears the branch with room, and the pairalpha_1 < 1/3and(1 + alpha_1) 2 beta < 1 - alpha_1, which a reading of the first two through the plainl^1exponent produces, is not a pair of separate demands. - Nothing in the bilinear half is base-10 mathematics. Refuted. No step of it needs
q = 10numerically, and no step of it needs the coefficient side past1-boundedness. The lattice and line sections never see the polytope, the coefficients or the sequence weighting the frequencies; those are spent one section earlier and survive only as the two window numbers. Base 10 enters in exactly three places and all three are names rather than arithmetic: the set of integers all of whose primes divide the base, the coprimality to the base, and dyadic parameters that are powers of the base. Every printed exponent in those sections is a rounding ofalpha,alpha_1orbeta:1/21rounds1 - alphaup,10/21roundsalpha/2down,27/77isalpha_1,50/77is1 - alpha_1,9/8rounds1 - alpha_1 + alpha/2down,3/16roundsalpha/2 - betadown,17/40is1 - 2 beta,9/25rounds(5/4) betaup, and23/80isbeta. - The criterion. Conjecture. A digit set satisfying (E1) whose
l^1exponent obeysalpha_1 < 1/4hassum_{n in S_F, n <= x, (n,q) = 1} mu(n) = O_B(A_F(x) (log x)^(-B))for everyB. This is a program, not a theorem, and its parts are named. The two steps that are aboutmurather than about primes are the two Proved statements above. The lattice and line estimates, the geometry of numbers and the exceptional-set bookkeeping are set-only or coefficient-free and transfer as read. The lattice branch, which was the one named gap, is closed by the two Proved bullets above. Three things are owed in its place and none of them is a new idea: the line branch at general base, whose two lemmas are set-free and coefficient-free but whose own conditionsm_t < (2 - t) betaandN >= x^max((5/4) beta, (5 beta - 1/2)/3)are not gathered into a statement anywhere here; and the write-out at general base of the two bookkeeping steps between the window and the pair geometry, the Parseval count of large frequencies for the Heath-Brown pieces and the dyadic reduction of the bilinear sum to the pair sum, both stated at source for arbitrary1-bounded sequences. The hypothesis(n, q) = 1is exactly what keeps every Type I modulus coprime to the base, since a Heath-Brown factorisation of an integer coprime toqhas every factor coprime toq; dropping it needs the level of distribution at base-divisible moduli, whose main term the bullet above prices and does not cancel. - Base 10 fails on one number, the moment. Proved. The threshold is
beta = 23/80 = 0.2875against1/4, a miss of3/80, which asksm_t < (2 - t)/4at sometin[1, 2)and at the source's ownt = 235/154asksm_t < 73/616 = 0.118506against the published59/433 = 0.136258, a required drop of4735/36344 = 0.130282of the value, more than the numerical slack in the eigenvalue bound that produces it. The greedy conditionalpha_1 + (5/2) beta <= 1asks onlybeta <= 20/77 = 0.259740there, so the threshold1/4is what binds and the miss is3/80exactly. A second miss is Refuted. Reading thel^1exponent against1/3substitutes the plainl^1exponent for the modulus exponent of the hybrid bound; with the true modulus exponent1 - alpha = 0.045757the condition is2 alpha_1 < alpha, which base 10 clears at0.701299against0.954242, and the companion condition(2 - alpha) 2 beta < 1 - alpha_1clears at0.601311against0.649350. Every one of the five lattice conditions holds at base 10 as published, as it must. The threshold read here is the sharp one,beta = inf_t m_t/(2 - t) <= 1/4; the headlinealpha_1 < 1/4is itst = 1proxy and base 10 misses that proxy too. So the obstruction there is the exceptional-set threshold alone, and closing it would close the criterion at base 10. - The least base that clears, and the family floor. Verified. The least base carrying a one-missing-digit set with
alpha_1 < 1/4isq = 21missing the digit0, and the least base whose whole one-missing-digit family clears isq = 34; both are certified, with their brackets and the bases they beat, in coprime. - The shape the route can deliver, at best. Proved. The conclusion is a log saving and not a power, and the binding step is the level of distribution rather than the arcs: the major-arc lemma gives
exp(-c sqrt(log x))and the minor arcs give a power, while the Type I input gives(log x)^(-B). So this route decides whetherM_F(x) = o(A_F(x))on the columns it reaches and says nothing whatever about the exponenttheta(F)below. It reaches only the columns withk > sqrt(q), by thel^1floor above, and only under(n, q) = 1, by the split above: three limits stated rather than assumed.
The exponent, tagged honestly
- Conjecture. For every digit set
Fwith2 <= k <= q - 1whose digit gcd is squarefree,theta(F) = 1/2: square-root cancellation against the set's own mass, the RH shape transplanted to the sparse column. The census is consistent with this and proves none of it: the 47 running-maximum exponents sit in[0.4465, 0.5358]with per-family drifts of0.0157..0.1056over the last five levels, and the full-set controls - whose limiting exponent is1/2under RH - read0.4413..0.4517at the same depths. A slope is a fit; the exact integers above are the claim, the exponent is not. - The believable refutation targets are one family with a proved exponent below
1/2(excess cancellation) or a proved omega-result (a family whose meter provably tracks its mass). The scaling mechanism produces neither: the vanishing family{0,4}atq = 5is total cancellation for the trivial reason4 | n, and its reduced column{0,1}carries the open question unchanged. - The technology gap is real: distribution of digit-restricted sets in residue classes is Erdos, Mauduit and Sarkozy 1998, the ellipsephic almost-primes rest on it (Dartyge and Mauduit 2000), and primes in one-excluded-digit sets took the full circle method at large base in Maynard 2019, whose Type I input (Proposition 7.1) is the set's own level of distribution in base 10: moduli coprime to 10 up to
X^(50/77), the residue0, a saving of any power oflog X, with50/77 = 1 - 27/77for thel^1exponent27/77of the digit transform (Lemma 10.3), itself the Markov eigenvalue boundlambda_(1,4) < 2.24190 < 10^(27/77)of that paper's (10.5). Among Maynard 2019, Maynard 2022, Nath 2024 and the sources REFS.md lists, that proposition is the one Type I statement for such a set, and it is stated for base 10 with one excluded digit: the general-base multi-digit version is a substitution sketched in its Section 16, reachings <= q^(23/80)excluded digits ands <= q - q^(57/80)when they are consecutive, and Maynard 2022 reachess < q^(1/5 - eps)andq - s >= q^(4/5 + eps)through the four Fourier norms of its Section 5 and the sketch of its Section 9. On the prime side Nath 2024 proves Bombieri-Vinogradov theorems forLambda(n) 1_A(n)at large base: unweighted with a maximum over residues only to levelX^(1/3 - delta), and nearX^(1/2)only against well-factorable weights, never unweighted; the set enters that proof through four norms of its transform (l^1, large sieve, hybrid,l^infinity) and never through a progression count, its only set-level fact being the count of the set in one last-digit class. Leng and Sawhney 2025 settle ternary Goldbach on the one-missing-digit set with thel^1boundg^(eps k)of the digit transform. The nearest multiplicative function computed over a missing-digit set is the divisor function (Kim 2024), and a provedthetafor any restricted column sits at or beyond that frontier; a whole-text search of the three circle-method sources finds the wordMobiusonce, as an inversion step inside the proof of Proposition 7.1,Liouvillenowhere, andMertensonly as Mertens' theorem on a product over primes, so none of them carries a Mobius or Mertens sum over the set. What the divisor section above adds against that is a power saving inxwhere Proposition 7.1 saves a power oflog X, uniform over up toq^(1 - eps)/2excluded digits, on the modulus ranged <= q^(1 - eps): sharper in saving type and in digit count, far shorter in range, and not a first level-of-distribution statement for such sets. Verified at source for Maynard 2019, Maynard 2022, Nath 2024 and Leng and Sawhney 2025. The census stands as the falsifiable record the eventual theorem must match.
Generators
- lab/mobius-designs prints every row, identity check, slope, distribution and band above:
CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p mobius-designs. - lab/rho-decoupling prints every divisor-section number:
CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p rho-decoupling. - lab/mertens-numerology prints every constant, table row, margin, rung and cost-out number of the GRH section:
CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p mertens-numerology; its 29 tests pin every rendered row as a string, the sign change of the certificate at3689 -> 3690, the exhaustive sweep of3690..10^5with its smallest step, the kernel bound against the exact shifted-grid sum on a4001-point grid, each ladder wall below4 * 10^6against a scan fromq = 3, and the constants of the general-bfloor bound. - The pair route has no generator of its own and needs none: every number in it is either exact rational arithmetic carried out in the sentence that prints it, an exponent quoted from the source named there, a certified base threshold whose generator is named on coprime, or a finite list that the sentence's own definition unrolls and a reader checks by hand.
- The Mertens control on the farey page is rendered by
lab/mertens-meter; the checkpoint controls here are the same function read at powers of the base.