first-base-below-a-quarter.md

Fix a base and cross out one digit; the integers that can still be written form a thin set with no multiplicative structure, and everything analytic known about it passes through the l^1 exponent alpha_1 of its digit transform, how much of the transform survives when the whole grid of frequencies is added up in absolute value. A quarter is a different bar: under the generalized Riemann hypothesis, expanding the Mobius sum over the set in additive characters and paying the uniform bound x^(3/4+eps) at each frequency beats the set's own counting function exactly when alpha_1 < 1/4, and Theorem 5.1 proves that transfer, |M_F(x)| << A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha. Base 21 with the digit 0 removed is the least base at which a one-missing-digit set is certified below the bar, alpha_1 in [0.2499765, 0.2499771]; bases 20, 10, 4 and 3 are certified above it at every digit, and the 83 sets of the bases 5 to 19 outside that list are swept by no generator here, which is the exact sense of "least". Base 34 is the least base from which every excluded digit clears, its 17 sets and the 3663 sets of the bases 35 to 125 certified and base 33 still missing at the digit 15, and an elementary majorant that forgets the digit proves alpha_1 < 1/4 from base 125 on. Every certified number is interval arithmetic with directed rounding, its generator named.

Introduction

Write every whole number in base q and throw away the ones that use the digit a_0: base 10 missing 7 leaves 9^L of them below 10^L, a set of dimension 0.954. Maynard (2019) proved that infinitely many of them are prime in base 10, reading the set through its transform hat F(t) = (1/(q-1)) sum_(a != a_0) e(at), e(x) = exp(2 pi i x), whose products over t, qt, ..., q^(L-1) t are the transforms of the L-digit strings. A sieve reads how much of that product survives when the q^L frequencies of the grid are added up in absolute value, and the growth rate of that sum, the l^1 exponent alpha_1 of Definition 2.5, is the number this paper is about; the published values sit near a third.

The figure is the census: one column per base from 10 to 40, one dash per distinct set at an upper bound on its exponent, the line at 1/4. The columns sink left to right, the spread inside a column is the excluded digit, cheapest at the ends and dearest near the middle, one dash crosses the line at base 21, the digit 0, and from base 34 on whole columns are under it. The bar is worth crossing because of what it buys: on the Mobius page the Mobius sum M_F(x) = sum_(n in S_F, n <= x) mu(n) is expanded in additive characters modulo q^L, at the cost of the l^1 mass of the transform, each frequency carries an exponential sum which under the generalized Riemann hypothesis is O(x^(3/4+eps)) uniformly by Baker and Harman (1991), and the two costs beat A_F(x) exactly when alpha_1 < 1/4, Theorem 5.1: every dash under the line is, under the hypothesis, a power saving for a Mobius sum.

Three results say where the dashes cross. The first is a theorem: alpha_1 < 1/4 for every base q >= 125 and every excluded digit, with the explicit bound of Theorem 4.7, uniform in the digit because its first step throws the digit away: the triangle inequality leaves a majorant built from the Dirichlet kernel, its level product expands over subsets of the digit positions, every maximal run telescopes into one Dirichlet kernel at a higher modulus, and what remains is a sum of Lebesgue sums, each bounded by Lemma 4.4. The other two are computations by a transfer matrix on windows of digits and the test of Collatz and Wielandt: base 21 missing 0 below the bar and base 20 above it at every digit (Facts 6.1 and 6.2), base 34 below at every digit and 33 above at the digit 15 (Fact 6.3), every set of the bases 35 to 125 below (Fact 6.4).

The literature, as read here. The base-10 value behind Maynard (2019) is alpha_1 <= 27/77 = 0.35065, the eigenvalue bound of its display (10.5), a finite-window upper bound by Fact 6.6; Karwatowski (2022), not read at source and carried from Granville (2024) and the author's base-9 paper, carries the same shape of bound to every base q >= 10. Erdos, Mauduit and Sarkozy (1998) and Konyagin (2001) distribute such sets in residue classes, Maynard (2022), Nath (2024) and Leng and Sawhney (2025) consume the exponent for primes, and Chow, Varju and Yu (2024) read the complementary exponent of the unshifted sum against 1/2, their Proposition 2.4 giving alpha_1^0 < 1/2 for every base q >= 5 with one digit removed. None fixes the least base at which a bar is met, and none carries a Mobius sum over a missing-digit set, the nearest being the divisor function of Kim (2024).

Definitions

Definition 2.1 (design). Fix a base q >= 3 and a digit set F, a subset of {0, ..., q-1} with k = |F| >= 2; q is the base and k the fill of this tree's vocabulary, and the two letters stand for them throughout. The design S_F is the set of positive integers whose every base-q digit lies in F, alpha = log k/log q its dimension, A_F(x) = #{n in S_F : n <= x}. For L >= 1, D_L is the set of the k^L integers 0 <= n < q^L whose L padded digits lie in F: when 0 in F it is S_F below q^L with 0 added, and when 0 is not in F no string over F starts with a zero and S_F below q^L is the disjoint union of D_1, ..., D_L. The one-missing-digit set at the digit a_0 is F = {0, ..., q-1} less {a_0}, k = q - 1.

Lemma 2.2 (the mass). For x >= q, x^alpha/k^2 < A_F(x) < 2k x^alpha.

Proof. Let L = floor(log_q x) >= 1. At most k^j members of S_F have j digits, so A_F(x) <= sum_(j <= L+1) k^j < (k^2/(k-1)) k^L <= 2k q^(L alpha) <= 2k x^alpha. Some digit of F is nonzero; the k^(L-1) strings of exactly L digits with that leading digit lie in S_F below q^L <= x, and k^(L-1) = q^((L+1) alpha)/k^2 > x^alpha/k^2. □

Definition 2.3 (the transform). hat F(t) = (1/k) sum_(a in F) e(at) and hat F_L(t) = prod_(j < L) hat F(q^j t) = k^(-L) sum_(n in D_L) e(nt), the normalised transform of D_L; both are 1-periodic, |hat F| <= 1, and |hat F(1 - t)| = |hat F(t)|.

Lemma 2.4 (the closed form and the mirror). Let D_q(t) = sin(pi q t)/sin(pi t), q at integers. For the one-missing-digit set at a_0, (q-1) |hat F(t)| = |D_q(t) - e((a_0 - (q-1)/2) t)|, and |hat F| is unchanged by a_0 -> q - 1 - a_0, so a base carries ceil(q/2) distinct sets, the pair coinciding only at odd q and a_0 = (q-1)/2.

Proof. The geometric sum sum_(a < q) e(at) = e((q-1)t/2) D_q(t) less the excluded term, times the unimodular e(-(q-1)t/2); and a -> q - 1 - a carries the set missing a_0 to the set missing q - 1 - a_0 and the sum to e((q-1)t) times its conjugate. □

Definition 2.5 (the grid sums and the exponent). For N >= 1 and real x,

Sigma_N(x) = sum_(i < q^N) |hat F_N(x + i/q^N)| ,   S_N = max_x Sigma_N(x) ,   alpha_1 = lim_N (1/N) log_q S_N = inf_N (1/N) log_q S_N .

The limit exists by Lemma 2.6; the exponent alpha_1^0 of q^L int_0^1 |hat F_L|, the average of Sigma_L over one cell, is at most alpha_1.

Lemma 2.6 (subadditivity). Sigma_N is q^(-N)-periodic, S_(N+M) <= S_N S_M, and S_N is nondecreasing in N; hence (1/N) log_q S_N converges to its infimum.

Proof. Replacing x by x + q^(-N) permutes the summands. Write i = b + q^N a with b < q^N, a < q^M, and t = x + i q^(-(N+M)). Since hat F is 1-periodic, q^M t = q^M x + b q^(-N) (mod 1), so the last N factors of |hat F_(N+M)(t)| are the summand of Sigma_N(q^M x) indexed by b and the first M factors the summand of Sigma_M(x + b q^(-N-M)) indexed by a; summing over a first, Sigma_(N+M)(x) = sum_(b < q^N) |hat F_N(q^M x + b q^(-N))| Sigma_M(x + b q^(-N-M)), at most S_M Sigma_N(q^M x) <= S_N S_M and at least Sigma_N(q^M x) min_y Sigma_M(y). At any shift y, Parseval on Z/q^M gives sum_i |hat F_M(y + i/q^M)|^2 = q^M k^(-M) with every term at most 1, so Sigma_M(y) >= q^M k^(-M) >= 1 and S_(N+M) >= S_N. Fekete's lemma gives the limit. □

Lemma 2.7 (the l^1 floor). For every base and every digit set, Sigma_N(0) >= q^(N(1 - alpha)), hence alpha_1 >= 1 - alpha, which is 0.01602 at base 21 missing one digit.

Proof. The Parseval identity of Lemma 2.6 at y = 0, with sum_i |z_i| >= (sum_i |z_i|^2)/max_i |z_i| and max_i |z_i| = |hat F_N(0)| = 1. □

The window machine

Proposition 3.1 (the window machine). Fix n >= 2 and split the circle into the q^n cells I_w = [w q^(-n), (w+1) q^(-n)). Put G^+(w) = sup_(I_w) |hat F|, G^-(w) = inf_(I_w) |hat F|, and let M^(+-) be the nonnegative q^(n-1) x q^(n-1) matrices (M^(+-) y)(v) = sum_(c < q) G^(+-)(vq + c) y((vq + c) mod q^(n-1)). Then alpha_1 <= log_q rho(M^+), and alpha_1 >= log_q rho(M'') for the restriction M'' of M^- to any subset of states; and for any y > 0, M^+ y <= mu y componentwise gives rho(M^+) <= mu, while M'' y >= mu y gives rho(M'') >= mu.

Proof. Write the digits of a cell index w < q^l as w = (w_1 ... w_l), most significant first, so that I_w is the set of t whose expansion starts 0.w_1 ... w_l; for t in I_w and j + n <= l, q^j t reduced mod 1 lies in the level-n cell v_j(w) = (w_(j+1) ... w_(j+n)).

The upper bound. Take l = L. For t in I_w and j <= L - n the factor |hat F(q^j t)| is at most G^+(v_j(w)), the remaining n - 1 factors at most 1, so U_L = sum_(w < q^L) sup_(I_w) |hat F_L| <= sum_(w < q^L) prod_(j <= L-n) G^+(v_j(w)) = 1^T (M^+)^(L-n+1) 1, the identity being the definition of M^+: a state is a string of n - 1 digits, appending c forms the window vq + c with weight G^+(vq + c) and moves to (vq + c) mod q^(n-1). Since x + i q^(-L) lies in I_i for x in [0, q^(-L)), S_L <= U_L, and Gelfand's formula gives limsup U_L^(1/L) <= rho(M^+).

The lower bound. Take l = L + n - 1, so every j < L has j + n <= l and on I_w every factor of |hat F_L| is at least G^-(v_j(w)). Hence int_0^1 |hat F_L| >= sum_(w < q^l) q^(-l) inf_(I_w) |hat F_L| >= q^(-(L+n-1)) 1^T (M^-)^L 1, and q^L int_0^1 |hat F_L| <= S_L gives S_L >= q^(-(n-1)) 1^T (M^-)^L 1. The entries are nonnegative, so 1^T (M^-)^L 1 >= 1^T (M'')^L 1, and if y > 0 on S' with M'' y >= mu y then 1^T (M'')^L 1 >= mu^L (sum_(S') y)/(max_(S') y), whence alpha_1 >= log_q mu; the best mu is rho(M''), by the Perron vector of an irreducible matrix or a further restriction of a reducible one.

The tests. If y > 0 and M^+ y <= mu y then every entry of (M^+)^L is O(mu^L), so rho(M^+) <= mu; the lower test is the display above. □

The last sentence is the test of Collatz (1942) and Wielandt (1950): a float power iteration supplies y, one pass in interval arithmetic turns it into a rigorous bound, and a poor vector weakens the bound and never breaks it. A cell containing a zero of |hat F| has G^- = 0, and the generators here report such a cell as undecided rather than restrict. Cell suprema and infima are enclosed by a sub-scan of m points plus the Lipschitz correction Lambda/(2 m q^n), Lambda = (2 pi/k) sum_(a in F) a.

The digit-uniform bound

Throughout this section F is the one-missing-digit set at the digit a_0, and c_1 = 2/pi.

Lemma 4.1 (the digit-blind majorant). For every q >= 3, every excluded digit and every real t, |hat F(t)| <= u_q(t) := min(1, (|D_q(t)| + 1)/(q-1)), and the right side does not mention a_0.

Proof. The triangle inequality on Lemma 2.4, and |hat F| <= 1. □

The discarded phase makes the majorant strictly larger than |hat F| almost everywhere, so the uniform bound is weaker than the per-digit one by construction; Fact 6.7 measures the cost. Write v_q(t) = (|D_q(t)| + 1)/(q-1) >= u_q(t) and Sigma^v_N(x) = sum_(i < q^N) prod_(j < N) v_q(q^j (x + i/q^N)) >= Sigma_N(x). A subset E of {0, ..., N-1} splits uniquely into maximal runs of consecutive positions, (s, l) the run occupying s, ..., s+l-1.

Lemma 4.2 (the run identity). For every x, (q-1)^N Sigma^v_N(x) = sum_(E) sum_(i < q^N) prod_((s,l) run of E) |D_(q^l)(q^s (x + i/q^N))|.

Proof. Expand prod_(j < N) (|D_q(q^j t)| + 1) over the subsets E of positions at which the kernel is taken; inside a run, with t' = q^s t, prod_(r < l) sin(pi q^(r+1) t')/sin(pi q^r t') = D_(q^l)(t'), and distinct runs occupy disjoint blocks. □

Every term is a Lebesgue sum. Let L_M = max_x sum_(j < M) |D_M(x + j/M)| and lambda_l = L_(q^l)/q^l.

Lemma 4.3 (the peel). Let (s_1, l_1), ..., (s_r, l_r) be the maximal runs of E, s_1 < ... < s_r, and t_i = x + i q^(-N). Then sum_(i < q^N) prod_k |D_(q^(l_k))(q^(s_k) t_i)| <= q^N prod_k lambda_(l_k) for every real x.

Proof. First, for M >= l >= 1 and real y, sum_(c < q^M) |D_(q^l)(y + c/q^M)| <= q^M lambda_l: writing c = rho + q^(M-l) c' with rho < q^(M-l), c' < q^l, the inner sum over c' runs over q^l points spaced q^(-l) apart and is at most L_(q^l). Set a_s = i mod q^(N-s); by periodicity the k-th factor depends on i through a_(s_k) alone, a_(s_1) determines every a_(s_k), and the top s_1 digits of i enter no factor, so it suffices to prove sum_(a_(s_1) < q^(N-s_1)) prod_k |D_(q^(l_k))(q^(s_k) x + a_(s_k)/q^(N-s_k))| <= q^(N-s_1) prod_k lambda_(l_k). Induct on r: for r = 1 it is the first display with M = N - s_1 >= l_1; for r >= 2 write a_(s_1) = a_(s_2) + q^(N-s_2) w with w < q^(s_2-s_1), so the factors k >= 2 depend on a_(s_2) alone and the first is |D_(q^(l_1))(z + w/q^(s_2-s_1))| at a fixed z; runs are maximal, so s_2 - s_1 > l_1, the first display bounds the inner sum by q^(s_2-s_1) lambda_(l_1), and the induction hypothesis bounds the outer sum. □

Lemma 4.4 (the Lebesgue constant). Let M >= 2 and gamma' = (2/pi)(gamma + log(8/pi)) = 0.9625228..., gamma Euler's constant. Then (i) L_M is attained at the half offset, L_M = 2 sum_(m < ceil(M/2)) csc((2m+1) pi/(2M)) - (M mod 2); (ii) L_M <= M((2/pi) log M + gamma') + 2/pi; (iii) lambda_l <= c_1 l log q + gamma' + c_1 q^(-l) for every l >= 1 and q >= 2.

Proof. (i) Write x = theta/M. All M points x + j/M share |sin(pi M t)| = |sin(pi theta)| =: s, so with d_j the distance from x + j/M to Z the sum is s sum_j csc(pi d_j). Replacing theta by 1 - theta reflects the point set, so take theta in [0, 1/2]; theta = 0 gives M, below the half-offset value, since the pair at m = 0 alone contributes 2 csc(pi/(2M)) > 4M/pi > M. Let theta in (0, 1/2]. The distances are the pairs {(m + theta)/M, (m + 1 - theta)/M} for m < ceil(M/2) - [M odd], with the single distance ((M-1)/2 + theta)/M when M is odd. Put mu_m = pi (m + 1/2)/M and delta = pi (1/2 - theta)/M in [0, pi/(2M)), so the pair sits at mu_m -+ delta, s = cos(M delta), and csc(mu - delta) + csc(mu + delta) = 2 sin mu cos delta/(sin^2 mu - sin^2 delta) writes its contribution as 2 csc(mu_m) X_m with X_m = cos(M delta) cos delta sin^2 mu_m/(sin^2 mu_m - sin^2 delta). The claim X_m <= 1 reads sin^2 mu_m (1 - cos(M delta) cos delta) >= sin^2 delta, whose left side is nondecreasing in mu_m on (0, pi/2], so it is enough at m = 0: put u = mu_0 = pi/(2M) <= pi/4 and delta = ru with 0 <= r < 1, so M delta = pi r/2. Since 1 - cos(pi r/2) cos(ru) = 2 sin^2(ru/2) + 2 cos(ru) sin^2(pi r/4) and sin^2(ru) = 4 sin^2(ru/2) cos^2(ru/2), the claim is

sin^2 u cos(ru) sin^2(pi r/4) >= sin^2(ru/2) (cos^2 u + cos(ru)) .

The function x cot x decreases on (0, pi), its derivative being (sin 2x - 2x)/(2 sin^2 x), so f(r) = sin(pi r/4)/sin(ru/2), whose derivative has the sign of (pi/4) cot(pi r/4) - (u/2) cot(ru/2) = (1/r)[(pi r/4) cot(pi r/4) - (ru/2) cot(ru/2)] with ru/2 < pi r/4 since u < pi/2, decreases on (0, 1] and f(r) >= f(1) = 1/(sqrt 2 sin(u/2)); at r = 0 the claim is 0 >= 0. Hence sin^2 u cos(ru) sin^2(pi r/4) >= sin^2(ru/2) sin^2 u cos(ru)/(2 sin^2(u/2)) = sin^2(ru/2) (1 + cos u) cos(ru), and (1 + cos u) cos(ru) - cos^2 u - cos(ru) = cos u (cos(ru) - cos u) >= 0. So every pair is at most 2 csc(mu_m), with equality at delta = 0, and the odd leftover at pi/2 - delta contributes cos(M delta)/cos delta <= 1 since delta <= M delta < pi/2. At the half offset s = 1, the pairs are 2 csc((2m+1) pi/(2M)) and the leftover is csc(pi/2) = 1, which the formula counts once.

(ii) Split csc x = 1/x + g(x). The expansion g(x) = sum_(k >= 1) 2 (2^(2k-1) - 1) |B_(2k)| x^(2k-1)/(2k)! has nonnegative coefficients, so g is nonnegative, increasing and convex on (0, pi), int_0^(pi/2) g = [log tan(x/2) - log x]_0^(pi/2) = log(4/pi), and the midpoint rule underestimates a convex integral; two harmonic inequalities come from the convexity of 1/x: the trapezoid rule on [N, N+1] gives log(1 + 1/N) <= (1/2)(1/N + 1/(N+1)), so H_N - log N - 1/(2N) is nondecreasing, and the midpoint rule on [N + 1/2, N + 3/2] gives log((N + 3/2)/(N + 1/2)) >= 1/(N+1), so H_N - log(N + 1/2) is nonincreasing; both tend to gamma, whence H_N <= log N + gamma + 1/(2N) and H_N >= log(N + 1/2) + gamma for every N >= 1.

Let M = 2n be even. The points x_m = (2m+1) pi/(2M), m < n, are the midpoints of the n intervals of length pi/M partitioning (0, pi/2], so sum_(m < n) g(x_m) <= (M/pi) log(4/pi), while sum_(m < n) 1/x_m = (2M/pi)(H_(2n) - H_n/2) <= (2M/pi)((1/2)(log M + log 2 + gamma) + 1/(2M)); doubling, L_M <= (2M/pi)(log M + gamma + log(8/pi)) + 2/pi.

Let M = 2n + 1 be odd, n >= 1. By (i), L_M = 2 sum_(m < n) csc(x_m) + 1, the x_m now the midpoints of n intervals of length pi/M inside (0, pi/2), so sum_(m < n) g(x_m) <= (M/pi) log(4/pi) as before. With 2n = M - 1 and n + 1/2 = M/2 the harmonic inequalities give H_(2n) - H_n/2 <= log(M-1) + gamma + 1/(2(M-1)) - (1/2)(log(M/2) + gamma) <= (1/2)(log M + log 2 + gamma) - 1/M + 1/(2(M-1)), using log(M-1) <= log M - 1/M. Multiplying by 4M/pi, the two small terms give -4/pi + 2M/(pi(M-1)) = -2/pi + 2/(pi(M-1)), so with the g part and the leftover 1, L_M <= (2M/pi)(log M + gamma + log(8/pi)) + 1 - 2/pi + 2/(pi(M-1)). For M >= 5 the constant is at most 1 - 3/(2 pi) < 0.53 < 2/pi; at M = 3, L_3 = 2 csc(pi/6) + 1 = 5 against 3((2/pi) log 3 + gamma') + 2/pi > 5.62.

(iii) is (ii) at M = q^l divided by q^l. □

Lemma 4.5 (the count). Put A_0 = 1 and A_N = sum_(E) prod_(runs) lambda_l. Then A_N = A_(N-1) + sum_(l < N) lambda_l A_(N-1-l) + lambda_N for N >= 1, and if z > 1 is the unique root of z = 1 + sum_(l >= 1) lambda_l z^(-l) then limsup A_N^(1/N) <= z; moreover z is nondecreasing in the sequence (lambda_l).

Proof. Split by the first position: if 0 is not in E the rest contributes A_(N-1); if the initial run has length l < N, position l is absent and the rest contributes lambda_l A_(N-1-l); l = N contributes lambda_N. Let Lambda(x) = sum_l lambda_l x^l, of radius 1 by Lemma 4.4 and infinite at 1; the recursion gives A(x)(1 - x - x Lambda(x)) = 1 + Lambda(x) for the generating function. phi(z) = 1 + Lambda(1/z) decreases strictly from infinity to 1 on (1, infinity), so z - phi(z) has a unique zero, and phi increases pointwise with the lambda_l. For |x| < 1/z, |x + x Lambda(x)| <= |x|(1 + Lambda(|x|)) < z^(-1) phi(z) = 1, so A is analytic there, and its coefficients being nonnegative its radius is at least 1/z. □

Corollary 4.6. alpha_1 <= log_q(z q/(q-1)) with z as in Lemma 4.5, since Lemmas 4.1 to 4.3 give (q-1)^N Sigma_N(x) <= q^N A_N for every x.

Theorem 4.7 (the digit-uniform bound). For every base q >= 3 and every excluded digit a_0,

alpha_1(q, a_0) <= log_q (z_q q/(q-1)) ,   where z_q is the unique root z > 1 of   (z-1)^3 = c_1 (log q) z + gamma' (z-1) + c_1 (z-1)^2/(qz - 1) .

In particular alpha_1 < 1/4 for every q >= 125 and every excluded digit; the right side reads 0.249980 at q = 125, 0.224161 at 250, 0.182916 at 1250 and 0.108912 at 10^6.

Proof. By Lemma 4.4(iii), lambda_l <= Lambda_l := c_1 l log q + gamma' + c_1 q^(-l), and by the monotonicity of Lemma 4.5 it is enough to bound the root of z = 1 + sum_l Lambda_l z^(-l); summing the three series, sum_l Lambda_l z^(-l) = c_1 (log q) z/(z-1)^2 + gamma'/(z-1) + c_1/(qz - 1), and multiplying by (z-1)^2 gives the cubic, so Corollary 4.6 gives the first claim. For the second, alpha_1 < 1/4 follows from z_q < w_q := q^(1/4)(1 - 1/q). Since phi(z) = 1 + sum_l Lambda_l z^(-l) is strictly decreasing and meets the diagonal once, z_q < w if and only if phi(w) < w, which after multiplying by (w-1)^2 is

(w-1)^3 > c_1 (log q) w + gamma' (w-1) + c_1 (w-1)^2/(qw - 1) .   (4.1)

Fact 4.8 verifies (4.1) at w = w_q for every 125 <= q < 3000. For q >= 3000: at the root, (z_q - 1)^3 >= c_1 (log q) z_q > 1 once q >= 5, so z_q >= 2 and z_q <= 2(z_q - 1); with (z_q - 1)/(q z_q - 1) < 1/q the cubic gives (z_q - 1)^2 <= 2 c_1 log q + gamma' + c_1/q <= 2 c_1 log q + 0.97 for q >= 86, so z_q <= g(q) := 1 + sqrt(2 c_1 log q + 0.97). At q = 3000, log q < 8.01 gives 2 c_1 log q + 0.97 < 11.18 < 3.35^2 and g(3000) < 4.35, while 7.4^4 < 3000 gives w_3000 > 7.39; and w_q - g(q) increases from there, since d w_q/dq > (1/4) q^(-3/4) and d g/dq = c_1/(q sqrt(2 c_1 log q + 0.97)), so it suffices that (1/4) q^(1/4) sqrt(2 c_1 log q + 0.97) > c_1, which reads 6.1 > 0.64 at q = 3000 with an increasing left side. □

Fact 4.8 (the certificate). With gamma' rounded up to 0.9625229, (4.1) holds at w = w_q for every integer 125 <= q < 3000 in interval arithmetic at 120 bits, tightest margin 3.985 * 10^(-3) at q = 125, and fails at q = 124 with margin below -6.16 * 10^(-2); a float root search finds no failure on [125, 10^5), the four values in Theorem 4.7 are rounded up in the sixth place, and the input rounded to c_1 l log q + 0.97 closes at 126. The scan of L_M at ten moduli from 4 to 10^5, three of them odd, finds the maximum at the half offset every time and the excess L_M/M - (2/pi) log M falling from 0.965217 to 0.962523, under gamma' + (2/pi)/M throughout. Generator: lab/py/digit-uniform-bound, verbs threshold and lebesgue.

The Mertens bar

Write M_F(x) = sum_(n in S_F, n <= x) mu(n), with mu the Mobius function, and S_mu(x, theta) = sum_(n <= x) mu(n) e(n theta).

Theorem 5.1 (the Mertens bar). Assume the generalized Riemann hypothesis, that L(s, chi) has no zero in Re s > 1/2 for every Dirichlet character chi. Let F be any digit set at base q with k >= 2, dimension alpha and l^1 exponent alpha_1. Then for every eps > 0 and all x >= 2,

|M_F(x)| <<_(q,F,eps) x^(alpha + alpha_1 - 1/4 + eps) ,   equivalently   |M_F(x)| <<_(q,F,eps) A_F(x)^(1 - delta + eps)   with   delta = (1/4 - alpha_1)/alpha .

The constant depends on q, F and eps alone, and the saving is a power exactly when alpha_1 < 1/4.

Proof. Step 1, orthogonality. For 0 <= n < q^L, completeness of the characters modulo q^L gives 1_(D_L)(n) = q^(-L) sum_(a < q^L) k^L hat F_L(a/q^L) e(-na/q^L), so for y < q^L, |sum_(n in D_L, n <= y) mu(n)| <= q^(-L) k^L Sigma_L(0) max_theta |S_mu(y, theta)|. Step 2, the l^1 mass. For every e > alpha_1 there is N with S_N <= q^(N e); writing L = aN + b with b < N, Lemma 2.6 gives S_L <= S_N^a S_b <= S_N^(a+1) <= q^(e(L+N)), so Sigma_L(0) <<_e q^(L e). Step 3, the Mobius input, quoted. Under the hypothesis, max_theta |S_mu(y, theta)| <<_eps y^(3/4 + eps), the case a = 1/2 of Baker and Harman (1991), whose hypothesis is that L(s, chi) is zero-free in Re s > a for every character and whose maximum is over all real theta, restated at source in Zhang (2024) and Porritt (2018); the one step not derived here. Step 4, assembly. Let L = ceil(log_q(x+1)), so x < q^L <= 2qx. If 0 in F, S_F below x is D_L less {0} in [1, x], and steps 1 to 3 give |M_F(x)| << q^(L(alpha - 1 + e)) x^(3/4 + eps) << x^(alpha + e - 1/4 + eps), using q^L <= 2qx when alpha - 1 + e > 0 and q^L > x otherwise. If 0 is not in F, S_F below x is the disjoint union over l <= L of D_l in [1, x], each block bounded the same way with y = min(x, q^l - 1); by Lemma 2.7, alpha + e > 1, so the sum over l is geometric and dominated by its top term. Take e = alpha_1 + eps. Step 5, the yardstick. By Lemma 2.2, x^alpha and A_F(x) agree up to the constants k^2 and 2k for x >= q, so x^(alpha(1 - delta) + eps) <<_(q,F,eps) A_F(x)^(1 - delta + eps/alpha). □

The Mobius page proves the same theorem with a digit-free kernel bound in place of alpha_1, closing at one missing digit only from base 1499, or 1032 at an end digit; the certified exponent moves that wall to 34. Under a zero-free half plane Re s > a with a > 1/2 the bar falls below 1/4, by the tables of Baker and Harman (1991) and Zhang (2024), and no set here is certified under it.

Corollary 5.2 (where the bar is cleared). Under the generalized Riemann hypothesis, |M_F(x)| <<_(q,F,eps) A_F(x)^(1 - delta + eps) with delta > 0 holds (i) at base 21 missing 0, with delta > 2.3 * 10^(-5); (ii) at every base q >= 34 and every excluded digit, with delta > 6.3 * 10^(-4) at q = 34; (iii) at every base q >= 125 and every excluded digit with delta >= (1/4 - u_q)/alpha, u_q the bound of Theorem 4.7, above 2.0 * 10^(-5) at 125.

Proof. Theorem 5.1 with Facts 6.1, 6.3 and 6.4 and Theorem 4.7: at base 21, 1/4 - alpha_1 > 2.29 * 10^(-5) and alpha < 0.984; at base 34, the digit 16 has 1/4 - alpha_1 > 6.299 * 10^(-4) and alpha < 0.9916; at 125, 1/4 - u_q >= 2.0 * 10^(-5). □

Proposition 5.3 (nothing unconditional). For any B with max_theta |S_mu(y, theta)| <= B(y), steps 1, 2 and 4 return nothing better than |M_F(x)| << B(x), since step 1 pays q^(-L) k^L Sigma_L(0) >= 1 against B by Lemma 2.7; the unconditional B(y) = y (log y)^(-A) of Davenport, read at source in Porritt (2018), returns x (log x)^(-A), above A_F(x) by the power x^(1 - alpha).

The census

Every number below is interval arithmetic with directed rounding: the transform evaluated at multiples of 2 pi/Q, Q = 8 m q^n, from enclosures of the cosines at 96 bits, every operation rounded outward, the Perron root bounded by the test of Proposition 3.1 with the float Perron vector, lower bounds truncated down and upper bounds rounded up. (n, m) is the window length in digits and the sub-scan.

Fact 6.1 (the first base). Base 21 with the digit 0 removed satisfies alpha_1 in [0.2499765, 0.2499771] at (6, 8), clearing 1/4 by more than 2.2 * 10^(-5); the same cell reads [0.2499715, 0.2499821] at (5, 8) and [0.2498658, 0.2500871] at (4, 8), which straddles the bar, and a second implementation returns [0.2499715, 0.2499822] at (5, 8). Generators: lab/py/digit-transform-norms, verbs six and least; lab/py/mobius-region, verb params 21 123456789abcdefghijk.

Fact 6.2 (what is certified below it). Base 20 satisfies alpha_1 > 0.2528608 at every one of its ten distinct sets at (4, 8), best at the digit 0 with [0.2528608, 0.2531118]. Base 10 satisfies alpha_1 > 1/4 at every one of its ten excluded digits, five distinct sets: the exceptional-set threshold beta of the pair route is certified above 1/4 at each, lower bounds 0.2502716 to 0.2541480, and beta <= alpha_1 since the t = 1 term of the infimum defining beta is the exponent of one shift of the grid sum. Bases 3 and 4 at every set and base 5 at the digits 0 and 1 are refuted by the census of the same route, whose region alpha_1 < alpha/2, beta <= min(1/4, (2/5)(1 - alpha_1)) has alpha/2 > 1/4 at those bases, so every refutation forces alpha_1 > 1/4; base 5 missing 2 is the one cell it cannot bracket from below. Table 1 certifies the digit 0 and the middle digit of bases 14 and 18. The remaining 83 sets of the bases 5 to 19 are certified by no generator of this tree: 21 is the least base at which a set is certified below 1/4. Generators: lab/py/digit-transform-norms, verb least; lab/py/mobius-region, verbs threshold and criterion.

Fact 6.3 (the first family). All 17 distinct sets of base 34 satisfy alpha_1 < 0.2493701 at (4, 8), the worst at the digit 16 with [0.2493107, 0.2493701] and the best at the digit 0 with [0.2246400, 0.2247052]; base 33 missing 15 satisfies alpha_1 > 0.2506145 at (4, 8). Generator: lab/py/digit-transform-norms, verb least.

Fact 6.4 (the family closed). All 3663 distinct sets of the 91 bases 35 <= q <= 125 satisfy alpha_1 < 1/4, each base at the shortest window in {2, 3, 4} that clears at sub-scan 8: bases 35 to 57 need three digits and every q >= 58 clears at two, the tightest row being base 58 missing 28 at alpha_1 < 0.2499305. Generator: lab/py/digit-transform-norms, verb family.

Corollary 6.5 (the family floor, in the half-line sense). Every base q >= 34 satisfies alpha_1(q, a_0) < 1/4 at every excluded digit, and 34 is the least base from which every base does, since 33 fails at the digit 15. Whether 34 is also the least single base whose every digit clears is not certified: of the bases 21 to 32, a digit above the bar is certified only at 22, 26 and 30.

Proof. Facts 6.3 and 6.4, Theorem 4.7 from 125, and the witness of Fact 6.3. □

Table 1. Certified brackets at (4, 8) at the cheapest and the dearest digit of ten bases. At odd q the middle digit is the constant-phase digit, where (q-1)|hat F(t)| = |D_q(t) - 1| has real zeros, a window cell empties and the machine prints the upper bound alone. Generator: lab/py/digit-transform-norms, verb least.

Fact 6.6 (the published exponent is an upper bound). Base 10 missing 5 satisfies alpha_1 in [0.3505775, 0.3505797] at (7, 16) with the wider box 2/10^n of the sieve literature, strictly below the published 27/77 = 0.3506494. Generator: lab/py/digit-transform-norms, verb six.

Fact 6.7 (the ceiling of the majorant). Run the window machine on the majorant u_q of Lemma 4.1 in place of |hat F|, cell suprema in closed form and rounded outward. The digit-blind bound clears 1/4 first at q = 75 at three window digits and at every base from 75 to 89; it reads 0.339085 at base 21 and 0.297296 at base 34 at four digits, against the per-digit 0.250088 and 0.249371 at the digits 0 and 16. So 75 is where the majorant stops working numerically and 125 where Theorem 4.7 proves it works; the exact crossing depends on the window and no value is claimed for it. Generator: lab/py/digit-uniform-bound, verbs least, window and compare, the per-digit column carried into compare from lab/py/digit-transform-norms.

The figure is the window machine on the supremum side run by its own binary, paper-first-base-below-a-quarter, over the 395 distinct sets of the bases 10 to 40 at windows of three to five digits and sub-scan 4: a dash per set at the float Collatz-Wielandt bound, 163 under the line. The binary asserts those counts, that no dash of the bases 10 to 20 is under the line, that base 21 has one, that each of 21 to 33 keeps one above and that every base from 34 has all below; float readings, not certificates.

What fails or is conditional

Four things. First, Theorem 5.1 is conditional and Proposition 5.3 says the same expansion returns nothing without the hypothesis; the saving at base 21 is 2.3 * 10^(-5) in the exponent, and the Mobius census on the Mobius page neither confirms nor threatens it. Second, the unconditional route is a program:

Conjecture 7.1 (the pair route). Let F be a digit set at base q such that every prime dividing the gcd of the differences of its digits divides q, with alpha_1 < 1/4. Then sum_(n in S_F, n <= x, gcd(n, q) = 1) mu(n) = O_B(A_F(x) (log x)^(-B)) for every B > 0.

The route follows the proof of Maynard (2019) with mu in place of the primes and is laid out on the Mobius page: the major arcs for mu at base-smooth moduli, free of exceptional zeros since every real primitive character of base-smooth modulus has conductor dividing 8 rad(q), and the level of distribution on an initial segment are proved there, the bilinear half never sees the coefficients, the region asked for is alpha_1 < alpha/2 and beta <= min(1/4, (2/5)(1 - alpha_1)), of which alpha_1 < 1/4 is the t = 1 proxy, and owed is the level of distribution at moduli sharing a factor with the base, which is why gcd(n, q) = 1 stays in the statement. The gate k >= q^(3/4), the Parseval floor beta >= 1 - alpha against beta <= 1/4, is met by every one-missing-digit set of base q >= 4, the conclusion would save a power of log x only, and the route is refuted at base 10 at every digit, where beta > 1/4 is certified. Third, the certificates have holes the machine reports: base 5 missing 2 and the middle digits of 21 and 33 carry an upper bound and no lower one, and the 83 sets of the bases 5 to 19 carry nothing. Fourth, the uniform bound is never the sharper number, Fact 6.7.

Open problems

Five things. The gap between 75 and 125: the largest loss in Section 4 is the peel, and treating the two-run terms directly should move 125 toward 75. The gap between 21 and 34: a lower bound at the middle digit uniform in q would replace the witness at 33 and settle the bases before it. The sweep below 20: the 83 sets of the bases 5 to 19 outside Fact 6.2 want one pass of the infimum machine each, the constant-phase digits with the restriction clause of Proposition 3.1, which the generators here do not implement. The exponent itself: every statement is a bracket, and whether alpha_1 is ever rational or algebraic is untouched. And whether the end digit is always the cheapest and the middle digit the dearest, as at every even base of Table 1 and at base 34 (base 33 puts digit 15 above digit 16), is not claimed.

Reproducibility

Three studies print every number of Sections 4 and 6, each run from the repository root with one verb and asserting every printed row. uv run python research/lab/py/digit-uniform-bound/ubound.py check in under half a minute covers the domination, the run identity, the peel, the Lebesgue scan, the threshold with its certificate and the consistency rows; its verbs threshold, lebesgue, least, window and compare, each under ten seconds, print Facts 4.8 and 6.7. uv run python research/lab/py/digit-transform-norms/norms.py least in 93 seconds prints Facts 6.2, 6.3, 6.6 and Table 1, six in 200 seconds Fact 6.6 and the six-digit bracket of Fact 6.1, family in 151 seconds Fact 6.4. uv run python research/lab/py/mobius-region/mobius_region.py params 21 123456789abcdefghijk in 12 seconds prints the second bracket of Fact 6.1, threshold in 41 seconds the base-10 refutation, criterion in 155 seconds the census of 49 designs. The figure is bash scripts/figures.sh paper-first-base-below-a-quarter, under three seconds a theme.

References

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