---
title: The First Certified Base Below a Quarter
lead: Base 21 missing the digit 0 is the least base at which a one-missing-digit set is certified below the `l^1` bar `1/4` of its digit transform, bases 3, 4, 10 and 20 certified above it at every digit and 83 sets of bases 5 to 19 unswept; every base from 34 clears at every digit, from 125 by a proof; under the generalized Riemann hypothesis each clearance is a power saving for the Mobius sum over the set.
date: 2026-09-21
figure: paper-first-base-below-a-quarter
shelf: first-base-below-a-quarter
---

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

![Every one-missing-digit set of the bases 10 to 40, one column per base and one dash per set at a float upper bound on its l^1 exponent from a window of three to five digits, 395 dashes in all; the quarter line crosses the picture, dashes above it dim, dashes below it yellow, the first yellow dash at base 21 and every dash of every column yellow from base 34 on.](paper-first-base-below-a-quarter)

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](../notes/mobius.md) 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](/wiki/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. □

| `q` | digit `0` | middle digit `floor(q/2)` |
| ---: | --- | --- |
| `10` | `[0.3090500, 0.3106411]` | `[0.3498936, 0.3512569]` |
| `14` | `[0.2780846, 0.2787259]` | `[0.3129308, 0.3134889]` |
| `18` | `[0.2596685, 0.2599987]` | `[0.2909812, 0.2912724]` |
| `20` | `[0.2528608, 0.2531118]` | `[0.2828721, 0.2830946]` |
| `21` | `[0.2498658, 0.2500871]` | `< 0.2666524` |
| `22` | `[0.2470967, 0.2472930]` | `[0.2760085, 0.2761833]` |
| `26` | `[0.2377912, 0.2379193]` | `[0.2649357, 0.2650507]` |
| `30` | `[0.2305272, 0.2306165]` | `[0.2563012, 0.2563819]` |
| `33` | `[0.2260074, 0.2260778]` | `< 0.2404222` |
| `34` | `[0.2246400, 0.2247052]` | `[0.2493107, 0.2493701]` |

**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](../notes/mobius.md) 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.

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