The digit-sum twist on a design

Fix a base b >= 3, a set E of m >= 1 excluded digits, the rest F with fill = b - m, and the missing-digit design S_F of mobius: the positive integers whose base-b digits all lie in F. That page bounds M_F(x) = sum_(n in S_F, n <= x) mu(n) unconditionally only at the shape A_F(x) exp(-c sqrt(log x)), and coprime carries the same shape for the primes of the design; in both, the ceiling is the major arcs, where mu and Lambda must be read in progressions mod powers of b through a zero-free region. This page twists the design by its digit sum and shows that the twisted transform loses a fixed share of its mass per pair of digits at every frequency, so the major arcs need no L-function and the saving becomes a power of x. Every number is printed by lab/py/digit-sum-twist.

The objects

s_b(n) is the sum of the base-b digits of n, D_k the digit strings of length k over F read as integers in [0, b^k), g_F(t) = sum_(d in F) e(d t) the digit symbol and A_F(x) the count of S_F up to x, all as on mobius.
The twisted weight is w(n) = 1_(S_F)(n) e(alpha s_b(n)) for a real alpha, and its transform at depth k is hat w_k(t) = sum_(n in D_k) e(alpha s_b(n) + n t). With n = sum_j d_j b^j the phase is sum_j d_j (b^j t + alpha), and the digits are independent, so hat w_k(t) = prod_(j < k) g_F(b^j t + alpha). At alpha = 0 this is the transform hat F_k of mobius.
The twisted meters are M_F(x; alpha) = sum_(n in S_F, n <= x) mu(n) e(alpha s_b(n)) and psi_F(x; alpha) = sum_(n in S_F, n <= x) Lambda(n) e(alpha s_b(n)).
beta = norm((b - 1) alpha) is the distance from (b - 1) alpha to the nearest integer, and the lattice is the set (b - 1)^(-1) Z of alpha with beta = 0.
B_b(F) = sup_t sum_(r mod b) abs(g_F((t + r)/b)) is the one-step constant of mobius, a supremum over every shift t. The certificate (W) of that page's unconditional dissection is B_b(F) <= fill b^(alpha_1) with alpha_1 < 1/5, read from the per-digit constants PB_b(m) or PB'_b(1, e_0); alpha_b = log(fill)/log(b).
V_F = fill^(-1) sum_(d in F) (d - mean(F))^2 is the variance of the digit set, K_b(x) = max(sin(pi b x)/sin(pi x), 1/sin(pi/b)) and x_f = 1/(2(b - 1)).

The two-position lemma

The lemma. Proved. Let F keep two consecutive digits, put x_0 = beta/(b + 1) and let G_F(x_0) = sup of abs(g_F(v)) over norm(v) >= x_0. Then for every real u and alpha, abs(g_F(u + alpha) g_F(b u + alpha)) <= fill G_F(x_0), and G_F(x_0) <= min(G_1, G_2, G_3) with the pair form G_1 = fill - 4 sin^2(pi x_0/2), the kernel form G_2 = m + K_b(x_0) and the variance form G_3 = max(fill sqrt(1 - 16 x_0^2 V_F), min(m + K_b(x_f), fill - 4 sin^2(pi x_f/2))). So with c = c(b, F, beta) = 1 - min(G_1, G_2, G_3)/fill, which is positive whenever beta > 0 because G_1 < fill, sup_t abs(hat w_k(t)) <= fill^k (1 - c)^(floor(k/2)) <= (1 - c)^(-1/2) fill^k b^(-gamma k) at every k, with gamma = log(1/(1 - c))/(2 log b).
Proof. Locating: put v_1 = u + alpha and v_2 = b u + alpha. Then b v_1 - v_2 = (b - 1) alpha, so beta = norm(b v_1 - v_2) <= b norm(v_1) + norm(v_2) <= (b + 1) max(norm(v_1), norm(v_2)), and one of the two phases has norm(v_i) >= x_0; its factor is at most G_F(x_0) and the other at most fill. The pair form: with d, d + 1 in F, abs(g_F(v)) <= fill - 2 + abs(1 + e(v)) = fill - 2 + 2 abs(cos(pi v)) = fill - 4 sin^2(pi norm(v)/2), which falls as norm(v) rises. The kernel form: g_F = D_b - g_E with D_b(v) = sum_(n < b) e(n v) and abs(g_E) <= m; at x = norm(v) in (0, 1/b], abs(D_b(v)) = sum_(n < b) cos(2 pi (n - (b - 1)/2) x), each term falling in x because 2 pi abs(n - (b - 1)/2) x < pi, so its supremum over [x_0, 1/b] sits at x_0 <= 1/(2(b + 1)) < 1/b, and on [1/b, 1/2] it is at most 1/sin(pi/b). The variance form: abs(g_F(v))^2 = sum_(d, d' in F) cos(2 pi (d - d') x) = fill^2 - 2 sum_(d, d') sin^2(pi (d - d') x), and at x <= x_f every abs(d - d') x <= 1/2, where sin^2(pi y) >= 4 y^2, while sum_(d, d') (d - d')^2 = 2 fill^2 V_F; so abs(g_F(v)) <= fill sqrt(1 - 16 x^2 V_F), falling in x, on [x_0, x_f], and past x_f, which is at most 1/b, the pair and kernel forms at x_f apply. The product: pair the positions 2i, 2i + 1 of hat w_k(t) = prod_(j < k) g_F(b^j t + alpha) with u = b^(2i) t, bound an odd last factor by fill, and use floor(k/2) >= (k - 1)/2.
The dichotomy is exact. Proved. On the lattice the lemma cannot hold: at t = -alpha every phase is b^j t + alpha = -(b^j - 1) alpha, an integer because b^j - 1 is a multiple of b - 1, so abs(hat w_k(-alpha)) = fill^k at every k. With two consecutive digits the twisted transform loses a fixed fraction of its mass at every frequency exactly when alpha is off the lattice.
The consecutive pair is needed. Verified. At base 5, F = {0, 2} and alpha = 1/8, off the lattice with beta = 1/2, every phase 5^j (3/8) + 1/8 = (3 * 5^j + 1)/8 lies in Z/2, where abs(g_F) = 2, so abs(hat w_k(3/8)) = 2^k = fill^k at every k; the generator reads 1 to twelve digits at k = 4, 8, 12 (verb lemma).
The constants. Verified. At base 10 missing 7: c >= 0.1500 and gamma >= 0.0352 at alpha = 1/2, c >= 0.0463 at 1/7, 0.0353 at 1/4, 0.0224 at 1/5, 0.0055 at 1/10, and 5.55 * 10^-5 at 1/900 (beta = 0.01), each from the variance form, which beats the pair form by a factor 66 at alpha = 1/2 and 61 at 1/900. At large base the kernel form carries beta = 1/2 and 0.2, c >= 0.3633 and 0.0644 at base 92317, and the variance form carries small beta, c >= 6.666 * 10^-7 at beta = 10^-3: to leading order it gives c = 8 x_0^2 V_F, and since V_F is about b^2/12 and x_0 about beta/b, that is c about (2/3) beta^2 at small beta and large base, where the pair form alone decays like beta^2 b^(-3) (verb lemma).

The twisted peel

The peel. Proved. For every alpha, every i >= 1 and every real s, the twisted grid sum Sigma_i(s) = sum_(a < b^i) abs(hat w_i(s + a/b^i)) obeys Sigma_i(s) <= B_b(F) Sigma_(i-1)(b s), so Sigma_i(s) <= B_b(F)^i, and under (W) Sigma_i(s) <= fill^i b^(i alpha_1): the certificate of mobius with C_F = 1, at every shift, for the twisted transform. Proof: hat w_i(t) = g_F(t + alpha) hat w_(i-1)(b t) and hat w_(i-1) has period 1. Write a = a' + r b^(i-1) with a' < b^(i-1) and r < b; then b (s + a/b^i) = b s + a'/b^(i-1) + r, so the second factor depends on a' alone, and s + a/b^i + alpha = (T + r)/b with T = b (s + alpha) + a'/b^(i-1), so the sum over r is sum_(r mod b) abs(g_F((T + r)/b)) <= B_b(F). The twist moves only the shift T, and B_b(F) is a supremum over every shift.
The hybrid bound transfers. Proved. The hybrid l^1 bound of mobius reads the transform through three things: the split hat F_k(t) = hat F_(i_1)(t) hat F_(k - i_1 - i_2)(V_1 t) hat F_(i_2)(y t/V_2), the certificate at a shift or at s = 0, and abs(g_F') <= pi b (b - 1) at one position. The split holds for hat w_k = prod_(j < k) g_F(b^j t + alpha) with the same three factors read for hat w, the derivative of g_F(v + alpha) in v has the same bound, and the shifted grids its proof integrates over are covered by the peel above, as is int_0^1 abs(hat w_i) <= b^(-i) B_b(F)^i. So under (W), sum_(a in R(D, H)) abs(hat w_k(a/y)) <= (1 + pi b) fill^k (V_1 V_2)^(alpha_1) at every alpha, with R(D, H), V_1 and V_2 as there.
The chain and the window do not transfer as written. The digit-uniform chain of coprime, which carries the untwisted certificate down to base 584, telescopes a run of positions into one Dirichlet kernel, prod_(j < l) D_b(b^j u) = D_(b^l)(u), which needs consecutive arguments in ratio b; the twisted arguments are b^j t + alpha, and b (b^j t + alpha) = b^(j+1) t + b alpha differs from b^(j+1) t + alpha mod 1 unless beta = 0. The window certificates of the same page bound a supremum over the one-parameter family of shifts s, while a twisted window sees the shift vector (b^j s + alpha)_j. Neither is refuted; neither is carried, so only the per-digit (W) serves here.
The readings. Verified. At base 10 missing 7 the grid supremum of sum_(r mod b) abs(g_F((T + r)/b)) reads 24.702183 on 2 * 10^5 shifts, certified below 24.702781 by the Lipschitz margin, against the floor 2(b - 1) = 18 and the step 3 bound 39.6623. Over 16 classes of alpha, 12 shifts and depths 1..5, 960 checks, the one-step identity holds to 1.40 * 10^-11 and Sigma_i(s)/B^i and Sigma_i(s)/(B Sigma_(i-1)(b s)) stay below 0.999914. The twist raises the unshifted mass, Sigma_5(0)^(1/5) reading 21.7151 at alpha = 1/2 and 22.3176 at 1/7 against 19.5656 untwisted: it moves mass toward the shifted supremum, which (W) already pays (verb peel).

The power saving

The theorem. Proved. Let F keep two consecutive digits and satisfy (W), so B_b(F) <= fill b^(alpha_1) with alpha_1 < 1/5, and let beta = norm((b - 1) alpha) > 0. With c = c(b, F, beta) and gamma of the two-position lemma, put sigma = gamma (1 - 4 alpha_1)/(5 - 4 alpha_1), eta = min(1/5 - alpha_1, sigma) and delta_0 = alpha_b eta/4. Then for every delta < delta_0 there is an effective C, depending on b, F, beta and delta alone, with abs(M_F(x; alpha)) <= C A_F(x) x^(-delta) and abs(psi_F(x; alpha)) <= C A_F(x) x^(-delta) at every x >= 2. Since c rises with beta, the same C serves every alpha with norm((b - 1) alpha) >= beta. The bullets below are the proof; the blocks, the dissection and the minor arcs are those of the unconditional dissection of mobius and its von Mangoldt form on coprime, and only the depth of the blocks and the major arcs change.
The blocks, at power depth. The block split of mobius writes S_F below x as at most fill + 1 blocks P y + D_k, y = b^k, at each scale k < L, the shorter elements included, and the point x. On a block s_b(P y + t) = s_b(P) + s_b(t) for t < y, so its twisted sum is e(alpha s_b(P)) Sigma(P, k) with Sigma(P, k) = sum_(t in D_k) e(alpha s_b(t)) mu(P y + t) = y^(-1) sum_(a mod y) hat w_k(a/y) S_P(-a/y), by orthogonality on Z/y applied to t -> 1_(D_k)(t) e(alpha s_b(t)), whose transform at a/y is hat w_k(a/y), and S_P(theta) = sum_(P y <= n < (P+1) y) mu(n) e(n theta); for Lambda read Lambda in both. Put tau = eta/4. The scales with b^k < x^(1 - tau) weigh at most 2 (fill + 1) fill^(k_0), k_0 the largest of them, with fill^(k_0) < x^((1 - tau) alpha_b), which is << A_F(x) x^(-tau alpha_b) by the mass floor A_F(x) >= fill^(L-1) - 1 >= (x/b)^(alpha_b) - 1, times log x for Lambda. Every other block has r = x/y <= x^tau.
The dissection. On such a block put Q = y^(3/5) and Z = y^zeta with zeta = 2 gamma/(5 - 4 alpha_1), which is below 2/5: each of G_1, G_2, G_3 is at least min(fill - 1, m + b/pi) >= fill/pi, so gamma <= log(pi)/(2 log 3) < 0.53 at every base, zeta < 0.26 and zeta < gamma/2; give each a mod y its Dirichlet fraction l/d, d <= Q, and its height h = abs(a d - l y) <= y^(2/5), and cut three regions: A is d >= y^(2/5), B is d < y^(2/5) with max(d, h) >= Z, and C is d < Z and h < Z, the union of the C1 and C2 of mobius.
The minor arcs. For mu the bullet The minor-arc input of mobius (Basak, Robles and Zaharescu, Theorem 1.4, at both approximations) and for Lambda the bullet The minor arcs, at source of coprime (Maynard 2022, Lemma 4.2, at a coarser fraction) give, with x^(o(1)) there replaced by the ratio r: on A, x d^(-1/2) <= x y^(-1/5) = x^(4/5) r^(1/5) and (x d)^(1/2) <= x^(4/5), so abs(S_P(a/y)) <<_eps x^(4/5 + eps) r^(1/5); on B the same term plus x (log x)^4 max(d, h)^(-1/2), the second approximation's x (h/y)^(1/2) <= x^(7/10) r^(3/10) lying below x^(4/5) r^(1/5) because r <= x.
Region A. By the peel at s = 0 its residues carry at most B_b(F)^k <= fill^k y^(alpha_1), so region A is <<_eps y^(-1) fill^k y^(alpha_1) (r y)^(4/5 + eps) r^(1/5) <= fill^k r^(1 + eps) y^(alpha_1 - 1/5 + eps).
Region B. Group its residues into the classes D <= d < 2D, H <= h < 2H or h = 0 of mobius, at most (log_2 y + 2)^2 of them, each with 16 b^2 D (D + H) <= 32 b^2 y^(4/5) <= y once y >= (32 b^2)^5 and max(D, H) > Z/2. By the transferred hybrid bound a class weighs at most y^(-1) abs(S_P) (1 + pi b) fill^k (32 b^2 max(D, H)^2)^(alpha_1). The first term of abs(S_P) with max(D, H) <= y^(2/5) gives <<_eps fill^k r^(1 + eps) y^((4/5) alpha_1 - 1/5 + eps), below region A's exponent, and the second with max(D, H) > Z/2 and 2 alpha_1 - 1/2 < 0 gives << fill^k r (log x)^4 Z^(2 alpha_1 - 1/2) = fill^k r (log x)^4 y^(-zeta (1/2 - 2 alpha_1)) = fill^k r (log x)^4 y^(-sigma).
Region C, with no L-function. It holds at most sum_(d < Z) phi(d) (2Z/d + 1) <= 3 Z^2 residues, abs(S_P) <= y for mu and <= y log(2x) for Lambda, and the two-position lemma gives abs(hat w_k(a/y)) <= (1 - c)^(-1/2) fill^k y^(-gamma) at every residue. So region C is at most 3 (1 - c)^(-1/2) fill^k Z^2 y^(-gamma) log(2x) = 3 (1 - c)^(-1/2) fill^k y^(2 zeta - gamma) log(2x), and gamma - 2 zeta = sigma by the choice of zeta. No Lemma A', no input (M), no Perron integral and no exceptional zero enters: the whole region, C1 and C2 together, is paid by the lemma.
Assembly. The three regions give abs(Sigma(P, k)) <<_eps fill^k r^(1 + eps) y^(-eta + eps) (log x)^6 with r <= x^tau and y >= x^(1 - tau), and the blocks at these scales, at most fill + 1 per scale, carry (fill + 1) sum_(k < L) fill^k <= (fill + 1) fill^L <= (fill + 1) fill (A_F(x) + 1). So they weigh <<_eps A_F(x) x^(tau (1 + eps) - (1 - tau)(eta - eps)) (log x)^6, and at tau = eta/4 the exponent is at most -3 eta/4 + eta^2/4 + 2 eps <= -eta/2 + 2 eps since eta <= 1/5; the discarded scales save x^(-tau alpha_b) = x^(-eta alpha_b/4). Every delta < alpha_b eta/4 = delta_0 follows on taking eps small, every step that asked y large holding for x past an effective point and the rest absorbed into C.
Where it holds. Proved. At one excluded digit (W) holds at every base >= 92317 through the step 3 constant, a half line by the proved floor of the ladder on mobius; so the theorem is Proved at every one-missing-digit set from base 92317. At m > 1 it holds wherever PB_b(m) < fill b^(-4/5), which admits m <= 176 at base 10^7; the wall itself is certified in interval arithmetic at 200 bits, 1/5 - alpha_1 >= 1.87123612 * 10^-8 at 92317 and <= -4.82092732 * 10^-8 at 92316 (verb lemma).
Below 92317 on the scan. Verified. On the float-scanned chord wall of mobius (W) holds from base 39363 at every excluded digit and from 28352 at e_0 in {0, b - 1}, so the theorem holds there on that scan, which is never Proved.
The exponent. Verified. delta_0 >= 4.678 * 10^-9 at base 92317 at beta = 1/2 and at beta = 0.2, where region A binds through 1/5 - alpha_1 >= 1.871 * 10^-8, and delta_0 >= 3.47 * 10^-10 at beta = 10^-3, where region C binds; on the chord wall delta_0 >= 6.741 * 10^-8 at base 39363, beta = 1/2; at base 10^6, delta_0 >= 2.819 * 10^-4 at beta = 1/2 and 4.163 * 10^-5 at beta = 0.2. Each is printed with c and gamma read at the extreme excluded digit, whose variance is the least, and floored at four digits. Region A binds only just above the wall: bisection between 92317 and 10^6 puts the switch to sigma at 100084 at beta = 1/2 and at 93409 at beta = 0.2, and sigma binds at 10^6 at both (verb lemma).
The paper. The theorem, the two-position lemma, the twisted peel and the corollary are written up for an outside reader as A Power Saving for the Digit Sum on Missing Digits.

Which region needs what

The ledger of inputs. Proved, by the proof above set beside the dissection of mobius.
Region A: the certificate (W) through the peel at s = 0 and the minor-arc exponent 4/5, unchanged by the twist; it asks alpha_1 < 1/5 and sets the wall.
Region B: (W) through the transferred hybrid bound and the second approximation, unchanged; it asks alpha_1 < 1/4, and with Z a power of y its saving Z^(2 alpha_1 - 1/2) becomes a power.
Region C, the old C1 and C2: only the two-position lemma, which asks two consecutive digits and beta > 0; in the untwisted dissection C1 needed Lemma A' of coprime and C2 needed the input (M) for mu or (M_Lambda) for Lambda, the zero-free regions of Chang and Martin and of Koukoulopoulos and the finite family of exceptional characters, and none of them is used.
The blocks: depth tau log_b x in place of the kappa sqrt(log x) of mobius, because no region now caps Z at exp(C_0 sqrt(log x)).
The certificate: only the per-digit (W) serves, so the theorem's wall at one excluded digit is that of (W), 92317 by proof and 39363 on the float scan, against 584, 301 and 115 for the untwisted bound on the chain and window certificates.
So the twist trades the base for the shape: the untwisted bound is Proved from base 584 and holds from base 115 on the window certificates, at exp(-c sqrt(log x)); the twisted one is Proved from base 92317, at x^(-delta).

The lattice and the ceiling

On the lattice the twist is a character. Proved. If (b - 1) alpha is an integer then e(alpha s_b(n)) = e(alpha n), because n - s_b(n) is a multiple of b - 1; so hat w_k(t) = hat F_k(t + alpha), the untwisted transform translated, and M_F(x; alpha) = sum_(c mod b-1) e(alpha c) M_F(x; b - 1, c), a combination of the untwisted meters in the progressions mod b - 1, M_F(x; b - 1, c) summing mu(n) over the n of S_F up to x with n = c mod b - 1. Read through the dissection, the major arcs of that sum sit at -alpha + l/b^j, where region C2 needs the input (M) at moduli dividing (b - 1) b^j: the zero-free region returns, with the primes of b - 1 added to the finite family of mobius, and the section Progressions and every modulus carries that case at exp(-c sqrt(log x)).
Where the ceiling sits. Proved. With two consecutive digits, sup_t abs(hat w_k(t)) = fill^k at every k when alpha is on the lattice and sup_t abs(hat w_k(t)) <= (1 - c)^(-1/2) fill^k b^(-gamma k) with c > 0 when it is off, by the dichotomy bullet. The mechanism is the phase map: the phases v_j = b^j t + alpha obey v_(j+1) = b v_j - (b - 1) alpha, full mass needs every v_j integral, and the integers are invariant under that affine map exactly when beta = 0. At alpha = 0 the frequencies of large mass are the b-adic preimages l/b^j of frequency 0, the C2 set, and the input of zero-free-region strength in the untwisted dissection is used at C2 and nowhere else. Off the lattice those preimages carry no mass, and the power saving holds at every alpha with beta >= beta_0 > 0, uniformly.
What the statement does not say. In the untwisted dissection C1 also caps the saving at exp(-c sqrt(log x)), through the rate c'^(floor(2k/(3 m_d))) of Lemma A' against the 3 Z^2 residues of the region, so the root-log ceiling there has two sources: the zero-free region, which sits exactly on the preimages of 0, and the sup-times-count pricing of C1, which needs no L-function. The twist removes both at once. Whether the untwisted C1 can be priced at a power is open here.

Primes of a design by digit sum

The corollary. Proved. Let F keep two consecutive digits and satisfy (W) in its per-digit form PB_b(m) < fill b^(-4/5), or at m = 1 its chord form, and let q >= 2 be coprime to b - 1. Then for every class c mod q and every delta below the delta_0 of the theorem at beta = 1/q, sum_(n in S_F, n <= x, s_b(n) = c mod q) Lambda(n) = psi_F(x)/q + O(A_F(x) x^(-delta)), and #{p in S_F : p <= x, s_b(p) = c mod q} = (1 + o(1)) pi_F(x)/q with pi_F(x) the count of primes of S_F up to x: the primes of a missing-digit design equidistribute in the classes of their digit sum, Gelfond's question asked on the design. Proof: orthogonality mod q writes the left side as q^(-1) sum_(r mod q) e(-r c/q) psi_F(x; r/q); the term r = 0 is psi_F(x)/q, and at r != 0 the number (b - 1) r/q is not an integer because gcd(q, b - 1) = 1 and q does not divide r, so norm((b - 1) r/q) >= 1/q and the theorem applies with beta = 1/q. The prime count of coprime, psi_F(x) = kappa_F A_F(x) + O(A_F(x) exp(-c sqrt(log x))), holds under (W), and kappa_F > 0 because the per-digit form of (W) forces m < b^(2/5) on mobius while phi(b) >= sqrt(b) at b > 6, so F holds a digit coprime to b. The bare inequality B_b(F) <= fill b^(alpha_1) does not suffice here: at b = 6^15 the digits sharing a factor with 6 keep the pair 2, 3 and satisfy it through the triangle inequality on three Dirichlet kernels, yet hold no digit coprime to b, so their design holds no prime past 3; the prime powers of S_F weigh at most x^(1/2) (log x)^2, below A_F(x) x^(-delta) since alpha_b > 1/2 + delta, and partial summation turns Lambda into the prime count.
The other moduli. When g = gcd(q, b - 1) > 1 the class of s_b(p) mod g is the class of p mod g, so the count asks for the primes of S_F in progressions mod g, the lattice case above; Mauduit and Rivat carry that case for all primes, with unrestricted digits, in their Theorem 3, and the bullet Gelfond at every modulus below carries it on the design from base 92317, with the error exp(-c sqrt(log x)).
The readings at base 10. Verified. Base 10 lies far below the wall of (W), so these readings test the shape and not the theorem. Below 10^7 the design missing 7 holds 266823 primes; their digit sums deviate from 1/q by at most 0.00825, 0.00880 and 0.00503 of the class size at q = 2, 4, 5, but by 0.05348, 0.10263 and 0.24272 at q = 7, 8, 10. The deviation is the design's and not the primes': the elements of the design coprime to b (b - 1) = 90, which carry the same mod 3 structure as the primes because s_b(n) = n mod 3, deviate by 0.04846, 0.09911 and 0.23939 on the same moduli, and the primes sit within 0.02213 of that model at every q coprime to 9. The design missing 0 reads the same way, 397866 primes, deviations 0.08164, 0.14497, 0.29072 at q = 7, 8, 10 against the model's 0.07811, 0.13695, 0.29155, primes within 0.02584 of the model. The slow modulus is the one with a small twist near the lattice: at q = 10 the class r = 3 sits at alpha = 3/10 = 1/3 - 1/30, and the twist -1/30, far from the lattice in beta = 0.3 but small in norm(alpha), mixes by abs(g_F(1/30))/fill per digit. At q = 3 the class 0 holds one prime, 3 itself (verb gelfond).

The falsification

The supremum, read. Verified. At base 10 missing 7 the supremum sup_t abs(hat w_k(t))/fill^k is bracketed at k = 1..8 by a max-product pass over the windows of six base-b digits of t, each window's factor bounded above by its centre value plus a Lipschitz margin, and bounded below by the value attained at the best path refined on a local grid; sixteen classes of alpha are sampled, five on the lattice and eleven off it. Off the lattice the attained value, the upper bound and the proved bound (1 - c)^(floor(k/2)) read 0.071152, 0.071165 and 0.521973 at k = 8, alpha = 1/2; 0.310870, 0.310915, 0.826973 at 1/7; 0.411592, 0.411649, 0.866052 at 1/4; 0.870069, 0.870167, 0.977921 at 1/10 and 0.870069, 0.870172, 0.977921 at 1/90; and 0.998609, 0.998716, 0.999778 at 1/900, where beta = 0.01; at every one of the eleven classes and every k >= 2 the upper bound sits below the proved bound, and the generator raises if either bracket fails to. On the lattice, at alpha = 0, 1/9, 1/3, 2/3, 4/9, the value at t = -alpha is 1 to twelve digits at k = 8: no decay, as the dichotomy demands. The class 1/3 is on the lattice at base 10, 9/3 = 3, so it must not decay and does not; the off-lattice classes with large beta, 1/27 with beta = 1/3 and 1/2 with beta = 1/2, decay. The measured rate is far better than the proved one: the upper bound's eighth root is 0.718676 at alpha = 1/2 against the proved (1 - c)^(1/2) = 0.921947, since the pairing books one deficit per two positions where the readings show one per position. At base 37 missing 18, k = 1..4, the same holds at its nine off-lattice classes, and the classes 1/2 and 1/4, on the lattice there because 36 is divisible by 4, read 1 (verb sup).
The twisted meters below 10^7. Verified. On the designs missing 7 and missing 0 at base 10, abs(M_F(x; alpha))/A_F(x) reads at most 0.0005 at x = 10^7 over all sixteen classes, and its running maximum at most 0.0006 of A_F(10^7), with no visible difference between the lattice and the rest: at this height mu cancels at every alpha, and a power x^(-delta) with delta near 10^-8 is invisible. The von Mangoldt meter separates them. On the lattice psi_F(10^7; alpha)/psi_F(10^7) sits within 0.0013 of the Ramanujan value c_q(r)/phi(q) that the primes carry mod q = 9/gcd(9 alpha, 9): -0.5000 at alpha = 1/3 and 2/3, near 0 at 1/9 and 4/9, where c_9(1) = 0. Off the lattice it sits within 0.0076 of the average of e(alpha s_b(n)) over the design's elements coprime to 90, the model of the corollary's readings, which is near 0 at most classes and is still large at 1/27, 1/90 and 1/900, reading 0.17, 0.86 and 0.998 in modulus on the design missing 7: those classes carry a small twist, and at 10^7 the frequency-zero term (g_F(alpha)/fill)^7 has barely begun to decay (verb meter).
Decay at alpha = 1/3 at base 10. Refuted. The expectation that the twisted supremum at base 10 decays at alpha = 1/3 is refuted by the witness abs(hat w_k(-1/3)) = fill^k at every k, since (b - 1)/3 = 3 puts 1/3 on the lattice; the twist it was meant to test is any alpha with 9 alpha not an integer, alpha = 1/2 the cleanest.

Against the literature

The digit sum of primes. Verified at source. Mauduit and Rivat 2010, read in the published text, prove in Theorem 1 that sum_(n <= x) Lambda(n) e(alpha s_q(n)) = O(x^(1 - sigma_q(alpha))) with sigma_q(alpha) > 0 whenever (q - 1) alpha is not an integer, and in Theorem 3 the count of primes with s_q(p) = a mod m, the main term (gcd(m, q - 1)/m) pi(x; a, gcd(m, q - 1)); the digits are unrestricted, mu is in none of their theorems, and their proof runs through Type I and Type II sums with a carry truncation, not through major arcs. Martin, Mauduit and Rivat 2014, read at source in the authors' copy, extend Theorem 1 to e(f(n) + beta n) for every digital function f(n) = sum_k alpha_k abs(n)_k with real alpha_k, with the explicit exponent c_q norm((q - 1) theta)^2 when the alpha_k form a progression, which they credit to Drmota, Mauduit and Rivat, and remark that mu follows by the same method; a real f makes e(f(n)) unimodular, so the indicator of a missing-digit set is outside that class.
The mechanism is in print for the full digit set. Verified at source. Maynard 2019 survey, read at source, proves a weak form of Mauduit-Rivat at large base by the circle method: its Lemma 8.4 (2) is the two-position lemma for the full digit set, abs(hat g(theta)) << b^(k (1 - delta_(alpha, b))) with delta_(alpha, b) = norm((b - 1) alpha)^2/(4 b^4), by the same locating step norm(b^i theta + alpha) + norm(b^(i+1) theta + alpha) >= norm((b - 1) alpha)/b, and its Lemma 10.1, as printed, bounds by that supremum times their count the major arcs whose denominator has a prime outside b, while its proof of Theorem 2.4 applies the same supremum-times-count to every small denominator, the mechanism of region C of this page; it treats the missing-digit set and the digit sum as two separate theorems and never their product. So the shape of this page's argument is in print; what is written here is its object: the twist carried onto a missing-digit design, for mu and Lambda, with an effective constant, from an explicit base, and a two-position constant that does not decay with b.
The missing-digit side. Maynard 2019 and Maynard 2022, as REFS.md records them, carry the primes of a missing-digit set and no digit-sum twist; the dissection of mobius came from them.
The search, stated as a search. A web search on the digit sum of primes with missing, restricted or ellipsephic digits, the publication lists of Martin and of Rivat, and the reference lists of Mauduit-Rivat 2010 and of the survey found no theorem on the digit sum of primes, or of mu, restricted to a missing-digit set; the nearest found are Aloui 2015, read at the publisher page in its abstract only; its definition of T_N(alpha, beta) and its Theorems 2.1, 2.5 and 2.6 are known here through a machine summary of that page and through the citation in Saavedra-Araya, the full text unread, which carries Gelfond's theorem onto the integers of a missing-digit set and their z-free members with power-saving errors through the twisted transform sum_(n in S_F, n < N) e(n theta + alpha s_b(n)), with no prime and no Mobius sum, the integer neighbour of the corollary here; digit sums of ellipsephic palindromes (Aloui, Mauduit and Mkaouar 2017), which are not primes, and Saavedra-Araya 2024, read at source in its abstract and Theorem A: for the integers of a missing-digit set and gcd(g, a) = gcd(a, a') = 1, the classes of n mod a and s_g(n) mod a' are jointly uniform exactly when gcd(a a', d_2 - d_1, ..., d_t - d_1) = 1, by Markov chains; integers only, no primes, the integer neighbour of the corollary here. This page is first in its object as far as that search reaches, and never in its shape.

Progressions and every modulus

Off the lattice the twist decays; on it the twist is a character of n mod b - 1, so the lattice case is the design in progressions. This section proves the Mobius and prime sums of S_F in every class mod N for every N coprime to b, at root-log strength, and combines them with the power saving into the law of s_b(p) mod q at every q.

The theorem in progressions. Proved. Let F keep two consecutive digits and carry the shifted-grid certificate of the unconditional dissection of mobius, sum_(a < b^i) abs(hat F_i(s + a/b^i)) <= C_F fill^i b^(i alpha_1) at every i and every real s, alpha_1 < 1/5, and let N >= 2 with gcd(N, b) = 1. Write M_F(x; N, r) and psi_F(x; N, r) for the sums of mu(n) and Lambda(n) over the n of S_F up to x with n = r mod N, and kappa_F = (b/phi(b)) #{f in F : gcd(f, b) = 1}/fill as on coprime. Then there are C, c > 0, effective and depending on b, F and N alone, with abs(M_F(x; N, r)) <= C A_F(x) exp(-c sqrt(log x)) at every r, and abs(psi_F(x; N, r) - kappa_F A_F(x)/phi(N)) <= C A_F(x) exp(-c sqrt(log x)) at every r with gcd(r, N) = 1, while at gcd(r, N) > 1 only powers of the primes of N count, at most omega(N) (log x)^2. At one missing digit the certificate holds from base 584 (mobius), so the theorem is Proved there at every N coprime to b, N = b - 1 among them; the bullets below are the proof.
Orthogonality and the blocks. M_F(x; N, r) = N^(-1) sum_(a mod N) e(-a r/N) M_F(x; a/N) with M_F(x; theta) = sum_(n in S_F, n <= x) mu(n) e(n theta). The block split of mobius is unchanged, and on a block e((P y + t) theta) = e(P y theta) e(t theta), so its sum is e(P y theta) y^(-1) sum_(a' mod y) hat F_k(theta + a'/y) S_P(-a'/y), by orthogonality on Z/y applied to t -> 1_(D_k)(t) e(t theta), whose transform at a'/y is hat F_k(theta + a'/y). S_P is the untwisted block sum, so the Dirichlet data l/d, h of a'/y, the cut into A, B and C, and the minor-arc bounds on S_P are those of the dissection verbatim; only the transform is translated by theta.
Regions A and B do not change. Region A reads the certificate at the shift theta in place of 0. The hybrid l^1 bound reads its top factor on the grid y theta/V_2 + a'/V_2, a shifted grid, and its bottom factor on the intervals theta + J_(l,d), still disjoint, through L^1 norms of hat F_(i_1) and its derivative over a period, which a translation does not change. Both bounds hold with the same constants, since the certificate holds at every shift.
Region C re-cut: the major arcs move to l/(N_a b^j). Put N_a = N/gcd(a, N). At a C residue a'/y = l/d + eta, d < Z, abs(eta) = h/(d y) < Z/y, the transform is read at a/N + l/d + eta. Write the reduced denominator of a/N + l/d as d''; then d'' < N Z. C1'' is the residues with d'' carrying a prime outside b: there the perturbed Lemma A' of coprime, as used in region C1 of mobius, gives abs(hat F_k(a/N + a'/y)) <= fill^k c'^(floor(2k/(3 m_(d'')))) with m_(d'') <= log_b(N Z) + 1, its hypothesis on eta unchanged, and the 3 Z^2 count and abs(S_P) <= y pay it exactly as C1, with log_b N added to m_d. C2'' is the rest: a/N + l/d = l''/d'' with d'' dividing a power of b, so l/d = l''/d'' - a/N and d divides d'' N_a, a divisor of (N b)^j. These are the frequencies where the translated transform can keep full mass: hat F_k at the b-adic fraction l''/d''. Partial summation of e(n eta) along the block gives abs(S_P(a'/y)) <= (1 + 2 pi h/d) d max_(u, r') 2 abs(M(u; d, r')) <= 16 Z max_(u <= x, r') abs(M(u; d, r')), whether or not d divides y, and C2'' holds at most 3 Z N'_Z residues, N'_Z <= (1 + log_2 Z)^(omega(N b)) the number of d < Z dividing a power of N b. So the one new input is (M) at the moduli d dividing a power of N b, the lemma below, and the assembly of mobius runs with c', c_1 and the block depth read at N b.
The lemma (M_N). Proved. Let R = N b. At every u <= x, every r and every d dividing a power of R with d <= exp(sqrt(log x)/2): abs(M(u; d, r)) <= C_1 x exp(-c_1 sqrt(log x)) and abs(psi(u; d, r) - 1_(gcd(r,d)=1) u/phi(d)) <= C_2 x exp(-c_2 sqrt(log x)), with C_i, c_i effective and depending on R alone. Proof. At u <= x exp(-sqrt(log x)) both are trivial. Otherwise put g = gcd(r, d); every n = r mod d has gcd(n, d) = g, so M = 0 unless g is squarefree, and then M(u; d, r) = mu(g) sum_c M(u/g; d_1, c) over the classes c mod d_1 = (d/g) s with c = r/g mod d/g and gcd(c, s) = 1, s the product of the primes of g not dividing d/g: at most rad(R) classes, d_1 <= d rad(R) dividing a power of R; for psi at g > 1 only powers of the primes of R count, at most omega(R) log u. Orthogonality bounds a class by the largest abs(M(v, chi)) over chi mod d_1, and 1/L(s, chi) = prod_(p) (1 - chi*(p) p^(-s))^(-1) / L(s, chi*) over the primes p of d_1 not dividing the conductor q* of the primitive chi* behind chi, so M(v, chi) = sum_w chi*(w) M(v/w, chi*) over the w built from those primes, at most (1 + log_2 v)^(omega(R)) of them; the terms with v/w < sqrt v are trivial and the rest have v/w >= x^(1/4) and q* <= rad(R) exp(sqrt(log x)/2). Chang and Martin 2019, read at source: Lemma 2.1 makes L(s, chi) zero-free in sigma >= 1 - c/log(q (abs(tau) + 4)) for every chi mod q, c effective, except possibly for one real zero of a single character mod q, with abs(log L) <= log log(q (abs(tau) + 4)) + O(1) in half that width for the others; Proposition 2.3 makes every L(s, chi mod q) zero-free in sigma >= 1 - c_0/(1 + log^+ abs(tau)), c_0 = eta/(q^(1/2) log^2 q), with abs(log L) <= (1/2) log q + 3 log log^+(q tau) + O(1) for the exceptional one; Lemma 2.4 turns a bound on abs(log L) into one on abs(1/L). The exceptional character is real, since a zero of a complex chi* is a zero of its conjugate, a second character mod q*; a real primitive character has conductor 2^e times an odd squarefree number with e in {0, 2, 3}, so here q* divides 8 rad(R), a finite family fixed by R, and c_0 >= eta/((8 rad(R))^(1/2) log^2(8 rad(R))). Perron's formula at 1 + 1/log v, height T = exp(sqrt(log v)), and the segment moved to 1 - (c/2)/log(q* (T + 4)), or to 1 - c_0/(1 + log T) for the finite family, give M(v, chi*) << v exp(-c_3 sqrt(log v)) with c_3 depending on R alone, since log(q* (T + 4)) <= 3 sqrt(log v); at q* = 1 the same argument runs on L(s, chi_0 mod 4) = (1 - 2^(-s)) zeta(s), which has no real zero in (0, 1) and so no exceptional zero. Summing back over w, c and g gives the bound on M. For psi the same Perron integral runs on -L'/L, bounded by Cauchy's estimate abs(L'/L) <= max abs(log L)/radius on discs that fit inside the regions where log L is bounded, so the segment sits at 1 - (c/4)/log(q* (T + 4)), a quarter of Lemma 2.1's width since its bound on log L holds in half of it, or at 1 - c_0/(2(1 + log T)), half of Proposition 2.3's width, for the finite family; Proposition 2.3's region excludes the exceptional zero itself, so no term v^(beta_1) appears, psi(v, chi*) << v exp(-c_3 sqrt(log v)) at q* > 1, the pole at q* = 1 gives v, and psi(v, chi) - psi(v, chi*) is at most omega(R) (log v)^2; orthogonality then gives psi(v; d_1, c) = v/phi(d_1) + O(v exp(-c_3 sqrt(log v))). Siegel's theorem is never used: the exceptional zeros lie on the real primitive characters of conductor dividing 8 rad(N b), the family of mobius with the primes of N added. The Mobius half is the proof of (M) on that page with rad(base) read as rad(N b) at each of its four uses: the classes c, the count of w, the conductor bound and the exceptional family; nothing else in it reads base. The von Mangoldt half is (M_Lambda) of coprime under the same swap, proved there through the exceptional term u^(beta_1) and its effective distance from 1, and here through the region of Proposition 2.3, which excludes it; the two routes agree.
The von Mangoldt main term, through a model. Proved. Put lambda_R(n) = (R/phi(R)) 1_(gcd(n, R) = 1). In each class mod d dividing a power of R, lambda_R sums to 1_(gcd(r,d)=1) u/phi(d) + O(2^(omega(R)) R), because n = r mod d coprime to R is a class mod d cut by the primes of R outside d, and (R/phi(R)) prod_(p | R, p not dividing d) (1 - 1/p) = d/phi(d); so Lambda - lambda_R obeys the bound of (M_N) for psi in every such class. Its exponential sum on a block is Lambda's, bounded on A and B by Maynard 2022, Lemma 4.2, as on coprime, less (R/phi(R)) sum_(e | R) mu(e) sum_(n in block, e | n) e(n theta), each inner sum at most min(y/e + 1, 1/(2 norm(e theta))): at a'/y = l/d + eta, norm(e a'/y) >= 1/(2d) when d does not divide e, as e abs(eta) <= R y^(-3/5)/d, which gives at most 2 d <= 2 y^(3/5), and when d divides e, so d <= R < Z and the residue is in B with h >= Z, norm(e a'/y) = e h/(d y) gives at most y/h; both lie below the minor-arc bounds of A and B. C1'' pays Lambda - lambda_R through abs(S_P) <= y (log 2x + R). So the dissection gives sum_(n in S_F, n <= x, n = r mod N) (Lambda(n) - lambda_R(n)) << A_F(x) exp(-c sqrt(log x)), and at gcd(r, N) = 1 the model is (R/phi(R)) #{n in S_F, n <= x : n = r mod N, gcd(n, b) = 1}, coprimality to N being automatic.
The local constants split evenly. Proved. On a block P y + D_k with k >= 1, gcd(n, b) = 1 reads the last digit, so with F* the digits of F coprime to b, #{t in D_k : gcd(t, b) = 1, P y + t = r mod N} = N^(-1) sum_(a mod N) e(-a (r - P y)/N) g_(F*)(a/N) prod_(1 <= j < k) g_F(a b^j/N). The term a = 0 is #F* fill^(k-1)/N. At a != 0, a b^j is nonzero mod N because gcd(b, N) = 1, so norm(a b^j/N) >= 1/N and the pair form of the two-position lemma gives abs(g_F(a b^j/N)) <= fill rho_N with rho_N = 1 - 4 sin^2(pi/(2N))/fill. Over the at most fill + 1 blocks per scale the errors sum to << (fill rho_N)^L << A_F(x) x^(-log(1/rho_N)/log b), and the blocks at scale 0 and the point x to O(b). So the count is (#F*/(fill N)) A_F(x) up to a power saving, and since R/phi(R) = (b/phi(b)) (N/phi(N)) the model is kappa_F A_F(x)/phi(N) + O(A_F(x) x^(-delta_N)). The design is uniform mod N in the limit whatever its digits share with N: the consecutive pair puts 1 among the digit differences, the condition of Saavedra-Araya 2024, Theorem A. At N = b - 1 the rate is exact: hat F_k(a/(b - 1)) = g_F(a/(b - 1))^k, since b^j = 1 mod b - 1.
Out of scope. gcd(N, b) > 1: then n mod gcd(N, b^j) is read on the last digits, the major arcs of the translated transform include denominators sharing primes with b in both factors, and this section does not carry it.
Gelfond at every modulus. Proved. Let F keep two consecutive digits and satisfy (W) in its per-digit form, so that the theorem of The power saving and the theorem above both hold, at one missing digit from base 92317. Let q >= 2, g = gcd(q, b - 1), N >= 1 coprime to b and N' = lcm(N, g). Then for every r_1 mod N and r_2 mod q, sum_(n in S_F, n <= x, n = r_1 mod N, s_b(n) = r_2 mod q) Lambda(n) = (g/q) kappa_F A_F(x)/phi(N') + O(A_F(x) exp(-c sqrt(log x))) when r_1 and r_2 agree mod gcd(N, g) and the class mod N' they define is prime to N', and O(A_F(x) exp(-c sqrt(log x))) otherwise, the implied constant and c effective and depending on b, F, N and q. At N = 1: s_b(p) mod q is uniform on the q phi(g)/g classes prime to g, the classes compatible with p mod g since s_b(p) = p mod g; at gcd(q, b - 1) = 1 this is the corollary above. Proof: orthogonality mod N and mod q writes the sum as (N q)^(-1) sum_(a mod N) sum_(c mod q) e(-a r_1/N - c r_2/q) sum_(n in S_F, n <= x) Lambda(n) e(a n/N + c s_b(n)/q). When q/g does not divide c, (b - 1) c/q is not an integer, since gcd(q/g, (b - 1)/g) = 1, so norm((b - 1) c/q) >= 1/q, and the theorem of The power saving with the frequency translated by a/N gives O(A_F(x) x^(-delta)): the two-position lemma bounds hat w_k at every frequency, the peel and the hybrid bound hold at every shift, and on a block e(a (P y + t)/N) = e(a P y/N) e(a t/N), so its proof runs with hat w_k(a/N + a'/y) and nothing else changes. When c = (q/g) c', e(c s_b(n)/q) = e(c' n/g) because g divides b - 1, and these terms sum to (g/q) psi_F(x; N', R') over the class R' mod N' with R' = r_1 mod N and R' = r_2 mod g, or to 0 when no such class exists; N' is coprime to b, and the theorem above gives the main term. The prime count follows by partial summation, as in the corollary: a prime dividing N' lies in no unit class mod N', and the prime powers of S_F weigh at most x^(1/2) (log x)^2.
The falsification. Verified. At base 10, below every wall and so a test of the shape, the primes of S_F above 30 and below 10^8, 2079504 missing 7 and 3103338 missing 0, read against the law in units of the square root of the predicted count: on the lattice classes, p mod 9, s_10(p) mod 3 and (p mod 7, s_10(p) mod 3), at most 0.83 and 0.76 from the law, and no prime in a class of law 0. The off-lattice part moves slower. The share of even s_10(p), which the law puts at 1/2, reads 0.48942, 0.49588 and 0.49464 below 10^6, 10^7 and 10^8 missing 7, and 0.48418, 0.49576 and 0.49144 missing 0, so s_10(p) mod 2, mod 6 and mod 18 drift from the law by up to 10.92 and 22.91 units at 10^8; the drift is not monotone in x, and it falls only at a fixed parity of the digit count. The drift is the prime 11 = b + 1 alone: on the 8-digit strings of the design the even share is 0.50000 prime to 30, 0.49583 prime to 330 and to 2310, 0.49996 prime to 390, and 0.49576 prime to 30030 missing 7, and 0.50000, 0.49228, 0.49228, 0.50000 and 0.49229 missing 0, so 7 and 13 add nothing, and the primes sit within 2.37 units of the strings prime to 30030 in every class tested. On the strings, the share prime to 330 less the share prime to 30 reads -0.00769, -0.00248, -0.00417, -0.00210 and -0.00103 at 6, 7, 8, 10 and 12 digits missing 7, and -0.01385, -0.00295, -0.00772, -0.00433 and -0.00243 missing 0: it oscillates with the parity of the digit count and, at even counts, falls by 0.54, 0.50, 0.49 and by 0.56, 0.56, 0.56 per two digits, against abs(g_F(1/22))^2/fill^2 = 0.480 and 0.561. That the drift is not in the limit is argued by the mechanism below and read in that decay. The mechanism is the two-position lemma's own extremal point: at alpha = 1/2 and the frequency l/(b + 1) with l = b/2, the phases b^j t + alpha alternate between +-1/(2(b + 1)), exactly the x_0 = beta/(b + 1) of the locating step, and the transform decays there only by abs(g_F(1/22))/fill, 0.6927 missing 7 and 0.7491 missing 0 per digit, against a prime weight mu(11)/phi(11) on that arc (verb lattice).
Against the literature. Verified at source. Nath 2024, read at source in its abstract and Theorems 1-3: for X = b^k, b sufficiently large in terms of delta (of order 10^632 at delta = 1/100 by its own remark), gcd(c, d) = gcd(r, b) = gcd(d, b) = 1, the error E(X; d, c; b, r) = sum_(n < X, n = c mod d, n = r mod b) Lambda(n) 1_A(n) - (1/phi(d)) (b/phi(b)) sum_(n < X, n = r mod b) 1_A(n) summed over d <= X^(1/3 - delta) with the max over c is << X^zeta (log X)^(-A). At one fixed d = N this is the prime count of the theorem above with its local constant kappa_F/phi(N) split evenly, at an unspecified large base, a log-power error and x a power of b; the lattice case for primes is in print in that form. What is here is the shape exp(-c sqrt(log x)) at every x, with an effective constant where that paper's depends on b and delta and is ineffective, through Bombieri-Vinogradov in its proof, from base 584, the Mobius sum, which that paper never names, and the digit-sum law at every q on a missing-digit set, which neither it nor Maynard 2019 (residue classes only through the set's Type I estimate, Proposition 7.1, moduli coprime to 10, as REFS.md records at source) carries. Mauduit and Rivat 2010, Theorem 3, is the unrestricted form of the law at N = 1, with the main term (gcd(q, b - 1)/q) pi(x; r_2, gcd(q, b - 1)), the same split.

Open

The pairing, which sets the exponent past about 10^5. The lemma books one deficit per two positions; the readings show close to one per position, 0.718676 per digit at alpha = 1/2 against the proved 0.921947. Since sigma binds from 100084 at beta = 1/2 and from 93409 at beta = 0.2 (the exponent bullet), the two-position constant, through regions B and C, limits delta_0 at almost every admissible base, and a deficit per position would raise it there.
A lower base. The wall is the per-digit (W). A twisted chain would need, in place of the Lebesgue sums of D_(b^l), the grid l^1 norms of the full-digit twisted kernel prod_(j < l) D_b(b^j t + alpha), the discrete transform Mauduit and Rivat estimate; a twisted window would need a supremum over shift vectors. Either could bring the theorem toward the untwisted 584 or 115.
The lattice at power strength. The section Progressions and every modulus carries every lattice alpha and every q at exp(-c sqrt(log x)), through (M_N) at the moduli N b^j; on the lattice the major arcs keep full mass, so a power saving there asks a zero-free strip for the L(s, chi) mod N b^j, and the error term of the law at every q stays root-log while the off-lattice part saves a power. The off-lattice rate at small base is set by the frequency l/(b + 1), where the locating step of the two-position lemma is attained.
The untwisted C1 at a power, and with it whether the root-log ceiling of mobius has any source besides the zero-free region at C2.
The exponent. Region A's 1/5 - alpha_1 binds only on a window just above 92317, and past it sigma, below gamma/4, binds; either way delta_0 stays below alpha_b/20, and nothing here touches the sparse columns or the exponent theta(F) of mobius.