half-interval-mobius.md

Write the integers in an odd base and keep those whose every digit is at most (base-1)/2, the lower half of the digits kept as an interval. They form a set S_F with A_F(x) = x^(alpha_base + o(1)) members up to x, where fill = (base+1)/2 and alpha_base = log fill/log base: thinner than the integers by a power of x, with no multiplicative structure, and at a prime base p exactly the set of k >= 1 for which p does not divide the central binomial coefficient C(2k,k). This paper proves two things on it with no hypothesis, at every odd base base >= 94939 and every x >= 2. The Mobius function cancels, |M_F(x)| <= C A_F(x) exp(-c sqrt(log x)), where M_F(x) sums mu(n) over the set up to x. And the primes are counted: the von Mangoldt sum over the set is kappa_F A_F(x) within the same error, kappa_F = base/(base+1), which is p/(p+1) at a prime base p. C and c are computable from the base alone. The proof is the dissection of the earlier paper An Unconditional Mertens Bound at Large Base, which asks the digit set for two consecutive digits and one number: a bound on the l^1 mass of the digit transform over every shifted grid, growing like fill^i base^(i alpha_1) with alpha_1 < 1/5. The earlier paper runs it for mu, and Section 7 runs it for Lambda. On the lower half the transform is a Dirichlet kernel of length fill, and a transfer operator with the explicit weight 1 + |sin(pi t)|/2 proves that bound with base^(alpha_1) = (2/pi) log base + O(1), the constant 2/pi sharp from both sides, where the proved one-step bound of the earlier paper pays sqrt(m) per digit at m avoided digits, a power of the base at the fill - 1 avoided here. The certificate clears 1/5 at every odd base from 94939, by 1.3678 * 10^(-8) at the wall, certified at 120 bits up to a monotone tail, and clears 1/4 from 3789, where under the generalized Riemann hypothesis it gives a power saving. The prime asymptotic on such sets at sufficiently large base, with a log-power error, is a theorem of Maynard (2022), which remarks at one missing digit that its error could be made effective; what is new here is the written proof of that effective shape on the half interval, each step that differs from the Mobius case written out, the Mobius side, and the explicit base.

Introduction

An unconditional Mertens bound on a digit set spends its base on one number. Expand the indicator of the strings of n digits over F in additive characters modulo base^n: every frequency a/base^n carries the Mobius exponential sum sum mu(v) e(va/base^n), weighted by the digit transform |hat F_n(a/base^n)|, and far from every fraction of small denominator that sum is bounded by x^(4/5 + eps), the minor-arc bound of Basak, Robles and Zaharescu (2023). Paid against the whole l^1 mass of the transform, that bound beats the count of the set exactly when the mass grows like fill^n base^(n alpha_1) with alpha_1 < 1/5. The figure plots, for the lower half F = {0, ..., (base-1)/2}, the margin 1/5 - alpha_1 of the certificate this paper proves, base^(alpha_1) = lambda/fill with lambda the closed form of Theorem 4.1. At small bases the mass is too large; the margin climbs like 1/5 - log((2/pi) log base)/log base and crosses zero at 94939 (Proposition 6.2). The orange curve is the same certificate against 1/4, the bar the generalized Riemann hypothesis asks, and crosses at 3789. Both curves are closed forms; the figure's binary recomputes them in floating point and asserts both walls and both margins against the generator, lab/py/interval-digits.

Theorem 1.1. Let base >= 94939 be odd, fill = (base+1)/2 and F = {0, 1, ..., fill-1}. Let S_F be the set of positive integers whose every base-base digit lies in F, A_F(x) = #{n in S_F : n <= x}, M_F(x) = sum_(n in S_F, n <= x) mu(n) and kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill. Then there are C > 0 and c > 0, depending on base alone and effectively computable, such that for every x >= 2

|M_F(x)| <= C A_F(x) exp(-c sqrt(log x)) ,
|sum_(n in S_F, n <= x) Lambda(n) - kappa_F A_F(x)| <= C A_F(x) exp(-c sqrt(log x)) .

Corollary 1.2 (the central binomial coefficients). Let p >= 94939 be prime and A(x) the number of integers 1 <= k <= x such that p does not divide C(2k,k). Then there are C > 0 and c > 0, depending on p alone and effectively computable, such that for every x >= 2

|sum_(1 <= k <= x, p does not divide C(2k,k)) mu(k)| <= C A(x) exp(-c sqrt(log x)) ,
|sum_(1 <= k <= x, p does not divide C(2k,k)) Lambda(k) - (p/(p+1)) A(x)|
    <= C A(x) exp(-c sqrt(log x)) .

Theorem 1.3 (under the generalized Riemann hypothesis). Assume that L(s, chi) has no zero in Re s > 1/2 for every Dirichlet character chi. Let base >= 3789 be odd and F, S_F, A_F, M_F as in Theorem 1.1, put alpha_base = log fill/log base, and let alpha_1 = log_base(lambda/fill) with lambda the explicit constant of Theorem 4.1. Then delta = (1/4 - alpha_1)/alpha_base > 0, and for every eps > 0 and every x >= 2, |M_F(x)| <<_(base, eps) A_F(x)^(1 - delta + eps). As the base grows, 1 - delta tends to 3/4.

The saving in Theorem 1.1 is of zero-free-region shape, not a power, and c is tiny; the constants are computable from the base but not printed, because the implied constant of the minor-arc input and the zero-free constants of the character bounds are unstated at their sources (Section 7). Below 94939 the theorem says nothing. The two bounds share the base 94939, the wall of the one certificate, and the dissection; they differ in the minor-arc input and the arithmetic input, so each has its own C and c, and Theorem 1.1 takes the larger C and the smaller c. The prime count adds the main-term constant kappa_F = base/(base + 1) (Lemma 7.3).

Why this set. The dissection of An Unconditional Mertens Bound at Large Base, called the earlier paper below, its Theorem 1.1, holds at any number of avoided digits and reads the digit set only through a shifted-grid l^1 certificate below 1/5 and two consecutive digits. The lower half keeps the pair {0, 1}, so everything rests on the certificate. There the proved one-step bound of the earlier paper, its Lemma 2.5, fails at every base: it is sqrt(m) + Phi/base per digit at m avoided digits, the sqrt(m) coming from Cauchy-Schwarz on the avoided digits, and at m = fill - 1 that term alone puts the exponent above 1/2 at every odd base from 5. The exact one-step constant max G is at most (3/2) lambda by Theorem 4.1, so it is logarithmic too, but a reading puts it a factor sqrt 2 above lambda in the leading term (Conjecture 9.2), and that factor is what the weight below buys. The lower half is also the set where the cost should be smallest, because its transform is a single Dirichlet kernel, |hat F(t)| = |sin(pi fill t)/sin(pi t)|, whose l^1 mass over one period is logarithmic in its length. The step from that heuristic to a certificate is a transfer operator T, Section 3, whose iterates are exactly the shifted grid sums, and a weight phi(t) = 1 + |sin(pi t)|/2 with T phi <= lambda phi everywhere, Theorem 4.1. The weight is chosen so that its extra term is summed in closed form: |hat F(u)| sin(pi u) = |sin(pi fill u)|, and fill is the inverse of 2 modulo the base, so the weighted sum over the grid is one geometric series (Lemma 3.3). The growth constant obeys lambda/fill <= (2/pi) log base + 2.60043004 + 3.4/base from base 101, and no shifted-grid certificate can do better than (2/pi) log base - 1.31 (Section 5); the bar asks lambda/fill < base^(1/5), and the closed form meets that power at 94939 (Section 6).

The prime asymptotic on the half interval at sufficiently large base, with a log-power error, is contained in Maynard (2022), Theorem 1.3, which takes consecutive excluded digits with q - s >= q^(4/5 + eps), and that paper remarks after its Theorem 1.1, at one missing digit, that Siegel zeros play no role at its moduli, so its errors could be made effective of the shape proved here, a remark its Section 9 does not restate for Theorem 1.3; Maynard (2019) gives the order of magnitude of the primes on such sets. Neither carries a Mobius sum. What this paper adds is the written proof of the effective shape on the half interval (Section 7), the Mobius bound, and the explicit base 94939 for both sums (Section 6), placed against the literature in Section 8.

Section 2 fixes the set and proves the binomial reading. Section 3 sets up the transfer operator, Section 4 proves the certificate, Section 5 the constant 2/pi from both sides, and Section 6 the walls with their margins. Section 7 restates what is reused from the earlier paper, proves the prime count by the same dissection with Lambda in place of mu, each changed step written out, and deduces Theorems 1.1 and 1.3 and Corollary 1.2. Section 8 places the result against the literature, Section 9 is the falsification, the limits and a conjecture on the true growth rate, and Section 10 says what is left open.

The half interval

Definition 2.1 (the half interval). Fix an odd base base >= 3 and put fill = (base+1)/2, so base = 2 fill - 1; the two words keep these meanings throughout, except in a statement made for any digit set, where fill is the number of its digits. The half interval is F = {0, 1, ..., fill-1}: m = fill - 1 avoided digits, all consecutive, the upper half. S_F, A_F(x) and M_F(x) are as in Theorem 1.1, and alpha_base = log fill/log base is the dimension of S_F. For n >= 0, D_n is the set of the fill^n integers 0 <= u < base^n whose n padded digits lie in F, so S_F below base^n is D_n less {0}. With e(t) = exp(2 pi i t), the digit transform is hat F(t) = sum_(a in F) e(at) and its level form is hat F_n(t) = prod_(i < n) hat F(base^i t) = sum_(u in D_n) e(ut), unnormalised as in the earlier paper, so |hat F_n| <= fill^n and

|hat F(t)| = |sin(pi fill t)/sin(pi t)| ,   read as fill at the integers .

|hat F| has period 1 and is even. Write ||z|| for the distance from z to the nearest integer. The set is the subject of the section The half interval of the note The Mobius meter across digit designs, which records every statement proved here.

Lemma 2.2 (the central binomial coefficients). At a prime base p, v_p(C(2k,k)) is the number of carries in the base-p addition k + k, so S_F = {k >= 1 : p does not divide C(2k,k)}.

Proof. Legendre's formula v_p(n!) = (n - s_p(n))/(p - 1), with s_p the base-p digit sum, gives v_p(C(2k,k)) = (2 s_p(k) - s_p(2k))/(p - 1). In the addition k + k each carry removes p from one position and adds 1 at the next, so s_p(2k) = 2 s_p(k) - (p - 1) c with c the number of carries, and v_p(C(2k,k)) = c. No carry enters the lowest position, so a first carry occurs at the lowest position whose digit d has 2d >= p; hence c = 0 exactly when every digit of k is at most (p-1)/2 = fill - 1. □

This is the case k + k of the theorem of Kummer (1852) that v_p(C(a + b, a)) counts the carries in a + b. So Corollary 1.2 is Theorem 1.1 read at a prime base, with A(x) = A_F(x).

Remark 2.3 (doubling). Every digit d of a member of S_F has 2d <= base - 1, so doubling carries nowhere and 2 S_F is the set of positive integers whose digits lie in the even digits {0, 2, ..., base-1}: a digit set with no two consecutive digits, all of whose members are even. Theorem 1.1 does not reach it; Section 10 says what is missing.

The transfer operator

Definition 3.1 (the operator and the grid sums). For any digit set F at the base base and any bounded phi of period 1, put

(T phi)(t) = sum_(r < base) |hat F((t+r)/base)| phi((t+r)/base) ,   G = T 1 ,
Sigma_i(s) = sum_(a < base^i) |hat F_i(s + a/base^i)| .

T is positive and keeps period 1: replacing t by t + 1 sends the term r to the term r + 1, and the term base - 1 to the term 0 moved by 1.

Lemma 3.2 (the transfer identity). For every digit set, every i >= 0, every phi of period 1 and every real t,

(T^i phi)(t) = sum_(a < base^i) |hat F_i((t+a)/base^i)| phi((t+a)/base^i) ,
so   Sigma_i(s) = (T^i 1)(base^i s) .

Hence if 1 <= phi <= Phi and T phi <= lambda phi everywhere, then Sigma_i(s) <= Phi lambda^i at every level i and every real shift s: a shifted-grid certificate in the sense of Theorem 1.1 of the earlier paper, with C_F = Phi and base^(alpha_1) = lambda/fill.

Proof. Induct on i, the case i = 0 being trivial. By the definition of T and the case i - 1 at the point (t + r)/base,

(T^i phi)(t) = sum_(r < base) |hat F((t+r)/base)| sum_(a' < base^(i-1))
                 |hat F_(i-1)((t + r + base a')/base^i)| phi((t + r + base a')/base^i) .

Put a = r + base a', which runs once over 0 <= a < base^i. Since hat F has period 1, hat F((t+r)/base) = hat F((t+a)/base) = hat F(base^(i-1) (t+a)/base^i), the factor at position i - 1 of hat F_i((t+a)/base^i), and hat F_(i-1)((t+a)/base^i) holds the positions below it; together they are hat F_i((t+a)/base^i). At phi = 1 and t = base^i s the right side is Sigma_i(s). T is positive, so T phi <= lambda phi gives T^i phi <= lambda^i phi by induction, and Sigma_i(s) = (T^i 1)(base^i s) <= (T^i phi)(base^i s) <= Phi lambda^i. The certificate of the earlier paper reads Sigma_i(s) <= C_F fill^i base^(i alpha_1), which is this with C_F = Phi and fill base^(alpha_1) = lambda. □

At phi = 1 the lemma is the peel of Lemma 2.3(i) of the earlier paper, with lambda = max G, the one-step constant B. A weight that is not constant lets a point where G is large borrow from points where phi is small; the certificate below spends that freedom on one sine.

Lemma 3.3 (the sine sum). On the half interval, with sigma the fractional part of fill t,

S(t) = sum_(r < base) |hat F((t+r)/base)| |sin(pi (t+r)/base)|
     = cos(pi (sigma - 1/2)/base)/sin(pi/(2 base)) <= csc(pi/(2 base)) .

Proof. |hat F(u)| |sin(pi u)| = |sin(pi fill u)| at every u, the integers included, so S(t) = sum_(r < base) |sin(pi (fill t + fill r)/base)|. Since 2 fill = base + 1, fill is the inverse of 2 modulo base, and fill r runs once over the residues modulo base as r does; |sin(pi z/base)| is unchanged when z moves by a multiple of base, and moving fill t to sigma by an integer permutes the residues again, so S(t) = sum_(j < base) sin(pi (sigma + j)/base), every argument in [0, pi). That is the imaginary part of e(sigma/(2 base)) sum_(j < base) e(j/(2 base)) = e(sigma/(2 base)) 2/(1 - e(1/(2 base))), and 2/(1 - e(1/(2 base))) = i e(-1/(4 base))/sin(pi/(2 base)). □

The certificate

Theorem 4.1 (the certificate). At every odd base base >= 3 put c_P = sqrt 2 - 4/pi, gamma Euler's constant,

X_0 = (base/pi)(log(base + 3) + gamma + log tan(3 pi/8 + pi/(4 base))) + c_P (base + 1)^2/(8 base) ,
lambda = fill + X_0 + csc(pi/(2 base))/2 ,                                               (4.1)

and phi(t) = 1 + |sin(pi t)|/2. Then T phi <= lambda phi everywhere on the half interval, so Sigma_i(s) <= (3/2) lambda^i at every level i and every real shift s: the half interval carries the shifted-grid certificate C_F = 3/2, base^(alpha_1) = lambda/fill.

Proof. T phi and lambda phi have period 1 and are even, because |hat F| and phi are and r -> base - 1 - r carries the points (1 - t + r)/base to 1 - (t + r)/base; so take t in [0, 1/2], and put a = sin(pi t/2) and b = cos(pi t/2). By Lemma 3.3, T phi = G + S/2 <= G + csc(pi/(2 base))/2, so it suffices to prove

G(t) <= fill + X_0 + sin(pi t) (X_0/2 + K_2 base/(4 pi)) ,                               (4.2)
K_2 = 2 (4/3 + (36/35)(7 zeta(3)/8 - 1)) < 2.7733 ,

since then T phi <= lambda + sin(pi t)(X_0/2 + K_2 base/(4 pi)) <= lambda + (lambda/2) sin(pi t) = lambda phi(t), the last step because K_2 base/(2 pi) < 0.45 base < fill.

The points. The base points u_r = (t + r)/base sit at distance delta = ||u_r|| from the integers: the low points r < fill at delta = (t + r)/base, and the high points r = base - 1 - r', 0 <= r' <= fill - 2, at delta = (1 - t + r')/base, every delta at most 1/2. There |hat F(u_r)| = |sin(pi fill delta)|/sin(pi delta), and fill delta = (t + r + delta)/2 at a low point and (1 - t + r' + delta)/2 at a high one. Expanding the sine of pi fill delta in the angles pi t/2 and pi delta/2 and dividing by sin(pi delta) = 2 sin(pi delta/2) cos(pi delta/2), every point weighs a combination of

P(delta) = 1/(2 sin(pi delta/2)) ,   Q(delta) = 1/(2 cos(pi delta/2)) ,

as in the table, with J = floor((fill-2)/2) and K = floor((fill-1)/2).

For instance at a low odd point pi fill delta = pi j + pi/2 + pi t/2 + pi delta/2, whose sine is cos(pi t/2 + pi delta/2) = b cos(pi delta/2) - a sin(pi delta/2) up to sign, and the angle pi t/2 + pi delta/2 is at most pi/2, so the weight is bP - aQ >= 0. At a high odd point pi fill delta = pi k + pi delta/2 - pi t/2, so the weight is abs(bQ - aP), which is bQ - aP where delta > t and at most aP where delta <= t. Every |hat F| is at most fill, which pays the point r = 0. So

G(t) <= fill + b Sigma_P + b Sigma_Q + a U + a Sigma_Q' ,

where Sigma_P sums P over the low odd and high even points, Sigma_Q sums Q over the low even and all the high odd points, U = sum_(low even) P + sum_(high odd, delta <= t) P - sum_(high odd, delta > t) P, and Sigma_Q' = sum_(j <= J) (Q((2j + 1 - t)/base) - Q((2j + 1 + t)/base)) collects the aQ terms; the high odd points with delta <= t are bounded by aP, their -bQ dropped.

The aQ terms. Q increases on [0, 1/2], and the high even point (2j + 1 - t)/base sits below the low odd point (2j + 1 + t)/base, so every term of Sigma_Q' is at most 0.

Two inequalities. On (0, 1/2], 1/(pi delta) <= P(delta) <= 1/(pi delta) + c_P delta: with z = pi delta/2 in (0, pi/4], 2 P(delta) - 2/(pi delta) = csc z - 1/z, which is positive, and convex because its Taylor series at 0 has positive coefficients, so it lies under its chord, csc z - 1/z <= (sqrt 2 - 4/pi)(4/pi) z = 2 c_P delta. And Q is convex on [0, 1), since sec is convex on [0, pi/2).

The bP terms. The low odd and high even points pair as (m + t)/base and (m - t)/base with m = 2j + 1, so

Sigma_P <= (base/pi) sum_(j <= J) (1/(m + t) + 1/(m - t)) + (c_P/base) sum_(j <= J) 2m ,
1/(m + t) + 1/(m - t) = 2/m + 2 t^2/(m (m^2 - t^2)) .

First, sum_(j <= J) 2/(2j + 1) = psi(J + 3/2) + gamma + 2 log 2 <= log(4J + 6) + gamma <= log(base + 3) + gamma, by psi(x) <= log x - 1/(2x) and 4J + 6 <= 2 fill + 2. Second, with t <= 1/2, 1/(1 - t^2) <= 4/3 at m = 1 and m^2/(m^2 - t^2) <= 36/35 at m >= 3, so the t^2 terms sum to at most 2 t^2 (4/3 + (36/35) sum_(m odd, m >= 3) m^(-3)) = K_2 t^2, the odd cubes summing to 7 zeta(3)/8 - 1. Third, sum_(j <= J) 2m = 2 (J + 1)^2 <= (base + 1)^2/8, since J + 1 <= fill/2. So Sigma_P <= (base/pi)(log(base + 3) + gamma + K_2 t^2) + c_P (base + 1)^2/(8 base).

The bQ terms. The low even points (2k + t)/base and the high odd points (2k - t)/base, 1 <= k <= K, are two progressions of spacing 2/base. By the Hermite-Hadamard inequality on the convex Q, Q(x) <= (base/2) int_(x - 1/base)^(x + 1/base) Q, and the intervals of one progression are disjoint and lie in [0, (2K + 1 + t)/base], inside [0, 1/2 + 1/base] because 2K + 1 <= fill. With int_0^y Q = (1/pi) log tan(pi/4 + pi y/4), Sigma_Q <= 2 (base/2)(1/pi) log tan(3 pi/8 + pi/(4 base)) = (base/pi) log tan(3 pi/8 + pi/(4 base)). So b Sigma_P + b Sigma_Q <= b X_0 + (base/pi) K_2 t^2.

The aP terms. This is the one step where the sign of the high odd weights is used. The low even points, delta = (2k + t)/base with 1 <= k <= K <= (base - 1)/4, have delta <= 1/2, so by the chord, 1/(2k + t) <= 1/(2k) and H_K <= log K + 1,

sum_(low even) P <= (base/pi) sum_(k <= K) 1/(2k) + c_P K/2
                 <= (base/(2 pi))(log((base - 1)/4) + 1) + c_P (base - 1)/8 ,

the right side positive also at K = 0, where the sum is empty. The high odd points, delta = (2k - t)/base, have delta <= t exactly when k <= X = t (base + 1)/2; put k* = min(floor(X), K). If k* = 0 every high odd point is subtracted and their signed sum is at most 0. Otherwise the points k <= k* carry at most (base/pi)(2/3 + (1/2) log((4k* - 1)/3)) + c_P X t: the term k = 1 has 1/(2 - t) <= 2/3, every k >= 2 has 1/(2k - t) <= 1/(2k - 1/2) <= (1/2) int_(k-1)^k dx/(x - 1/4), and the k* values of delta are each at most t, with k* <= X. The points k > k* all have delta > t and are subtracted, and they carry at least (base/(2 pi)) log((K + 1)/(k* + 1)), from P(delta) >= 1/(pi delta) and 1/(2k - t) >= 1/(2k) >= (1/2) int_k^(k+1) dx/x. Now (4k* - 1)(k* + 1) <= 4 X^2 + 3 X <= (base + 1)(base + 4)/4, since X <= (base + 1)/4, and K + 1 >= fill/2 = (base + 1)/4, so with X t <= (base + 1)/8 the signed high odd sum is at most

(base/pi)(2/3 + (1/2) log((base + 4)/3)) + c_P (base + 1)/8 .

Adding the low even bound,

U <= (base/pi)((1/2) log((base - 1)(base + 4)/12) + 7/6) + c_P base/4
  <= (base/pi)(log(base + 4) - 0.0757) + c_P base/4 ,

since (base - 1)(base + 4) <= (base + 4)^2 and 7/6 - (1/2) log 12 < -0.0757. Against X_0, with log tan(3 pi/8 + pi/(4 base)) >= log tan(3 pi/8) = log(1 + sqrt 2), gamma + log(1 + sqrt 2) > 1.4585, log((base + 3)/(base + 4)) >= -1/(base + 3) and (base + 1)^2/(8 base) >= 0,

X_0 - U >= (base/pi)(1.4585 + 0.0757 - 1/(base + 3)) - c_P base/4 > 0 ,

because (1.5342 - 1/6)/pi > 0.43 > c_P/4. So U < X_0, and a U <= a X_0, also when U < 0.

Assembly. G(t) <= fill + (a + b) X_0 + (base/pi) K_2 t^2. Since (1 - a)(1 - b) >= 0, a + b <= 1 + ab = 1 + sin(pi t)/2; since sin(pi t) >= 2t on [0, 1/2], t^2 <= t/2 <= sin(pi t)/4. That is (4.2). □

The proof uses the shape of the half interval at three points: |hat F| is one kernel sin(pi fill u)/sin(pi u), which gives the table; fill is the inverse of 2, which gives Lemma 3.3; and the weights at the high odd points change sign at delta = t, which the signed sum U follows. Fact 9.1 checks on grids of shifts the bounds on the harmonic parts of Sigma_P and U and on Sigma_Q, the closed form of Lemma 3.3, the bound (4.2) and T phi <= lambda phi, and on small grids the level sums against (3/2) lambda^i; the elementary inequalities between those steps are not checked separately. The bound (4.2) is tight to 1.5 * 10^(-4) of its size at t = 0 (lab/py/interval-digits, verb check).

The constant 2/pi

The growth constant of Theorem 4.1 is fill times (2/pi) log base + O(1), and no shifted-grid certificate does better in the leading term.

Proposition 5.1 (the upper constant). At every odd base base >= 101,

lambda/fill <= (2/pi) log base + c_inf + 3.4/base ,
c_inf = 1 + 2/pi + (2/pi)(gamma + log(1 + sqrt 2)) + c_P/4 <= 2.60043004 .

Proof. lambda/fill = 1 + 2 X_0/(base + 1) + csc(pi/(2 base))/(base + 1). In 2 X_0/(base + 1) the factor 2 base/(base + 1) < 2 multiplies (1/pi)(log(base + 3) + gamma + log tan(3 pi/8 + pi/(4 base))), which is positive; log(base + 3) <= log base + 3/base; log tan z has slope 2/sin(2z), below 2.9 on [3 pi/8, 3 pi/8 + pi/(4 base)] once base >= 101, so log tan(3 pi/8 + pi/(4 base)) <= log(1 + sqrt 2) + 2.9 pi/(4 base); and the last term of X_0 gives c_P (base + 1)/(4 base) = c_P/4 + c_P/(4 base). For the cosecant, csc z - 1/z is convex on (0, pi/2] and equals 1 - 2/pi at the end, so csc z <= 1/z + (2/pi)(1 - 2/pi) z and csc(pi/(2 base))/(base + 1) < 2/pi + (1 - 2/pi)/(base (base + 1)). Collecting, lambda/fill <= (2/pi) log base + c_inf + E/base with E = (2/pi)(3 + 2.9 pi/4) + c_P/4 + (1 - 2/pi)/(base + 1) < 3.399 at base >= 101. The constant c_inf is bounded at 120 bits, and the inequality is also checked directly, the closed form (4.1) against the right side at 120 bits, at every odd base 101..3001 and at 94939, 200001, 10^6 + 1 and 10^8 + 1, the smallest gap at least 1.3332 * 10^(-7) (lab/py/interval-digits, verb wall). □

Proposition 5.2 (the lower constant). At every odd base base >= 3 and every t, G(t) >= (base/pi) log((base + 2)/5) - fill/4 and G(t) >= base. At every odd base base >= 9, min G/fill >= (2/pi) log base - 1.31 > 0, so every grid sum, shifted or not, has Sigma_i(s) >= (((2/pi) log base - 1.31) fill)^i at every level i and shift s; and Sigma_i(s) >= base^i at every odd base.

Proof. Keep t in [0, 1/2] and the table of Theorem 4.1. Every weight is nonnegative, so drop the point r = 0, the high odd points and the bQ of the low even points. The low odd and high even points carry b (P((m + t)/base) + P((m - t)/base)) - a (Q((m + t)/base) - Q((m - t)/base)) at m = 2j + 1. With P(delta) >= 1/(pi delta) and 1/(m + t) + 1/(m - t) >= 2/m, the first part is at least b (base/pi) sum_(j <= J) 2/(2j + 1) >= b (base/pi)(log(base + 1) + gamma - 2/fill), by psi(x) >= log x - 1/x, 4J + 6 >= base + 1 and J + 3/2 >= fill/2. Q runs from 1/2 to 1/sqrt 2 on [0, 1/2], so each difference of Q lies in [0, 1/2], and there are J + 1 <= fill/2 of them, so with a <= 1 they cost at most fill/4. The low even points carry at least a (base/pi) sum_(k <= K) 1/(2k + 1/2) >= a (base/(2 pi)) log((4K + 5)/5) >= a (base/(2 pi)) log((base + 2)/5), by 1/(k + 1/4) >= int_k^(k+1) dx/(x + 1/4) and 4K + 5 >= base + 2. Since log(base + 1) + gamma - 2/fill >= log((base + 2)/5) >= 0 and b + a/2 >= 1 on [0, 1/2], the latter because cos z + (1/2) sin z is concave on [0, pi/4] and at least 1 at both ends, G(t) >= (base/pi) log((base + 2)/5) - fill/4. Parseval on Z/base gives sum_r |hat F((t+r)/base)|^2 = base fill at every t, and each term is at most fill, so G >= base. Dividing by fill, min G/fill >= (2 base/(pi (base + 1))) log((base + 2)/5) - 1/4. That is at least (2/pi) log base - 1.31 at every odd base 9..9999, certified at 120 bits, and at every base from 101 because the difference is at least 1.06 - (2/pi) log 5 - (2/pi) log((base + 2)/5)/(base + 1) >= 0.016518 there, the last term decreasing from base 11; the factor (2/pi) log base - 1.31 is positive from 9 and negative at 7 (lab/py/interval-digits, verb wall). Finally T^i 1 >= (min G)^i pointwise, because T is positive and T 1 = G, and Lemma 3.2 turns both lower bounds on G into lower bounds on Sigma_i(s). □

So the l^1 cost of the half interval per digit is (2/pi) log base + O(1) from both sides, in the normalisation base^(alpha_1) of the certificate: any shifted-grid certificate (C_F, alpha_1) has base^(alpha_1) >= (2/pi) log base - 1.31 at every odd base from 9, since its C_F is fixed while i grows, and the one of Theorem 4.1 has base^(alpha_1) <= (2/pi) log base + 2.64 from 101, by Proposition 5.1, since 2.60043004 + 3.4/101 < 2.6342. Measured against base, as the Parseval floor Sigma_i >= base^i measures it, the cost is (1/pi) log base per digit. The weight of Theorem 4.1 is what reaches the constant 2/pi: the constant weight phi = 1 pays the one-step constant max G, which reads sqrt 2 times larger in the leading term (Conjecture 9.2).

The walls

The bar 1/5 asks lambda/fill < base^(1/5), a logarithm against a power, and the power wins from a computable base on.

Lemma 6.1 (the tails climb). For e in {1/5, 1/4} put h_e(base) = base^e - (2/pi) log base - c_inf - 3.4/base. Then h_(1/5) increases on base >= 327 and h_(1/4) increases on base >= 43.

Proof. h_e'(base) = e base^(e-1) - 2/(pi base) + 3.4/base^2 > (e base^e - 2/pi)/base, which is positive once base^e > 2/(pi e), that is once base > (10/pi)^5 = 326.78... at e = 1/5 and base > (8/pi)^4 = 42.05... at e = 1/4. □

Proposition 6.2 (the walls). The certificate of Theorem 4.1 has alpha_1 < 1/5 at every odd base base >= 94939 and alpha_1 > 1/5 at 94937; it has alpha_1 < 1/4 at every odd base base >= 3789 and alpha_1 > 1/4 at 3787. At the walls and the bases below them,

and farther out alpha_1 <= 0.1993872 at 100003, 0.1761232 at 1000003 and 0.1331636 at 10^9 + 7.

Proof. alpha_1 < e is lambda/fill < base^e, and by Proposition 5.1 base^e - lambda/fill >= h_e(base) at every odd base >= 101. In interval arithmetic at 120 bits the least integer at which the tail h_(1/5) is certified positive is 94946, and for h_(1/4) it is 3793; both lie above the thresholds of Lemma 6.1, so h_e > 0 at every real base beyond, and the bar holds at every odd base from 94947 and from 3793. Below the tail, the closed form (4.1) is evaluated at 120 bits at every odd base 94939..94947 and 3789..3793, every gap base^e - lambda/fill certified positive, and at 94937 and 3787 the gap is certified negative, with the values in the table. Each bar is read as the exact interval 1/5 or 1/4, and bar - alpha_1 is printed from the interval of log(lambda/fill)/log base, never by differencing two rounded numbers; lower ends are truncated down and upper ends rounded up (lab/py/interval-digits, verb wall, under a second). □

The margin 1/5 - alpha_1 at the wall is thin, 1.3678 * 10^(-8), and it enters the proof of Theorem 1.1 only in region A: for mu through eps = (1/5 - alpha_1)/4, which fixes the implied constant of the minor-arc input, and for Lambda, which has no eps, as the power saving y^(alpha_1 - 1/5) of Section 7. Either way it fixes how large x must be before the power saving of region A takes over: it moves C, never c. The wall is the certificate's: it says where this phi clears the bar, not where the route stops (Conjecture 9.2).

The reduction to the dissection

The Mobius half of the dissection is reused unchanged. This section restates it, checks each place where the digit set enters, proves the constant kappa_F, and proves the prime count by the same dissection with Lambda in place of mu, each step that changes written out.

Theorem 7.1 (the dissection, restated). Let base >= 3 and let F be a set of fill digits with 2 <= fill < base, so that at least one digit is avoided, with S_F, A_F, M_F, D_n and Sigma_i as above. Suppose F contains two consecutive digits and carries a shifted-grid certificate below 1/5: constants C_F >= 1 and alpha_1 < 1/5 with Sigma_i(s) <= C_F fill^i base^(i alpha_1) at every i >= 0 and every real s. Then there are C > 0 and c > 0, depending on base and F alone and effectively computable, such that for every x >= 2

(i)   |M_F(x)| <= C A_F(x) exp(-c sqrt(log x)) ,
(ii)  |sum_(n in S_F, n <= x) Lambda(n) - kappa_F A_F(x)| <= C A_F(x) exp(-c sqrt(log x)) ,

with kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill. Part (i) is proved in the earlier paper. Part (ii) is proved below, as the subsection The dissection for the von Mangoldt function of the note The coprimality spine gives it: Lemma 7.2 carries its main term, and Lemmas 7.4 and 7.5, Proposition 7.6 and the proof after it carry the rest.

Part (i) is Theorem 1.1 of An Unconditional Mertens Bound at Large Base, stated there for any set of m >= 1 avoided digits, with the remark after its proof that no step uses m = 1. The proof cuts x into blocks Py + D_n with y = base^n within x^(o(1)) of x, expands each block in additive characters modulo y, and cuts the frequencies into four regions by Dirichlet approximation at Q = y^(3/5) and Z = exp(C_0 sqrt(log x)): the minor arcs A, the middle denominators B, the fractions C1 whose denominator has a prime outside the base, and the fractions C2 whose denominator divides a power of the base. Part (ii) runs the same proof with Lambda in place of mu and changes three inputs: the main term, which the principal characters of region C2 carry, Lemma 7.2; the minor-arc input, Lemma 4.2 of Maynard (2022) in place of Basak, Robles and Zaharescu (2023), Lemma 7.4; and the arithmetic input, primes in progressions to moduli dividing a power of the base, where a possible exceptional zero leaves a secondary term that a finite family of characters fixed by the base bounds, so that Siegel's theorem is never used, Lemma 7.5. Two trivial bounds gain a factor log x, since Lambda <= log x where |mu| <= 1. Everything else is identical, the hybrid l^1 lemma included: the earlier paper proves it under a certificate with the constant C_F^2 (1 + pi base^2 C_F/fill), and part (i) uses it in that form too.

The digit set enters that proof at six places, and the half interval meets each one.

• The certificate, at shift 0 in region A, where the whole l^1 mass is paid against the uniform x^(4/5 + eps), and at every shift in the hybrid l^1 lemma of region B, which asks only alpha_1 < 1/4. Theorem 4.1 and Proposition 6.2 supply it with C_F = 3/2 from 94939.
• Two consecutive digits, in region C1, through the contraction of |hat F_n| near a fraction with a denominator prime to the base, Lemma 6.1 of the earlier paper. The half interval keeps {0, 1}.
• fill, in the contraction constant rho = 1 - (2/fill)(1 - cos(pi/(4 base))) of that lemma, which the assembly carries into C_0 and c. It depends on the base alone.
• At most fill + 1 blocks at each scale in the block split, Lemma 3.2 of the earlier paper, at any number of avoided digits.
• The mass floor A_F(x) >= fill^(L-1) - 1 of the same lemma, which the half interval meets directly since 0 is in F.
• For the prime count only, kappa_F, read on the last digit, Lemma 7.2 and Lemma 7.3.

No large sieve and no hybrid estimate is imported: the hybrid l^1 bound, Lemma 5.1 of the earlier paper, is proved there from the certificate at every shift and |hat F'| <= pi base (base - 1), which holds at every digit set. What the proof imports, none of it reading the digit set: for mu, the minor-arc bound of Basak, Robles and Zaharescu (2023), Theorem 1.4, with Theorems 23.5 and 23.6 of Koukoulopoulos (2019) inside its proof, and for the characters of region C2 Exercise 8.4 and Theorem 7.2 of Koukoulopoulos (2019) with Lemma 2.1, Lemma 2.4 and Proposition 2.3 of Chang and Martin (2019); for Lambda, Lemma 4.2 of Maynard (2022) and Theorems 12.3, 12.4 and 12.8 of Koukoulopoulos (2019). Effective here means computable from the base, not explicit: the implied constant of the minor-arc bound at the eps used, the zero-free constants of the character bounds and the constants of the progression bounds are unstated at their sources, so C and c are computable and not printed. The bound of the earlier paper on the saving, c <= sqrt(|log rho|/24)/40 with |log rho| <= pi^2/(15 fill base^2), holds unchanged.

Lemma 7.2 (the main term). Let F be any digit set, n >= 1, y = base^n and P >= 0, and, with Lambda(0) = 0, write the block sum as sum_(u in D_n) Lambda(Py + u) = y^(-1) sum_(a mod y) hat F_n(a/y) S_P(-a/y) with S_P(theta) = sum_(Py <= v < (P+1)y) Lambda(v) e(v theta). At a residue a of region C2, a/y = l/d + j/y with (l, d) = 1, d < Z dividing a power of base and |j| d = h < Z, let the principal part of S_P(-a/y) be that sum with Lambda(v) replaced, on each class v = r mod d, by its mean (d/phi(d)) 1_(gcd(r,d)=1). If rad(base) < Z <= 2^n and rad(base) < Q, with Q = y^(3/5) the Dirichlet parameter of the dissection, the principal parts vanish unless j = 0 and d divides rad(base), and over region C2

y^(-1) sum_(a in C2) hat F_n(a/y) (principal part of S_P(-a/y))
    = (base/phi(base)) #{u in D_n : gcd(u, base) = 1} = kappa_F fill^n .

Proof. Every prime power p^e exactly dividing d has 2^e <= p^e < Z <= 2^n, so e < n and d divides y. On the class r mod d, 0 <= r < d, the block holds the y/d points v = Py + r + dw, 0 <= w < y/d, and e(-va/y) = e(-rl/d) e(-rj/y) e(-wj/(y/d)), because Pyl/d, Pj and wl are integers. The sum over w is y/d when y/d divides j and 0 otherwise, and |j| < Z/d < y/d, so only j = 0 survives, where the principal part is (d/phi(d))(y/d) sum_(gcd(r,d)=1) e(-rl/d) = y c_d(l)/phi(d) = y mu(d)/phi(d), c_d the Ramanujan sum, since (l, d) = 1. It vanishes unless d is squarefree, and a squarefree d dividing a power of base divides rad(base). Conversely every d dividing rad(base) is below Z, and each residue a = ly/d with (l, d) = 1 has l/d as its fraction, since a second fraction l'/d' with d' <= Q within 1/(d'Q) of it would force 1/(dd') <= 1/(d'Q), that is d >= Q, against d <= rad(base) < Q. So the left side is sum_(d | rad(base)) (mu(d)/phi(d)) sum_(gcd(l,d)=1) hat F_n(l/d) = sum_(u in D_n) sum_(d | rad(base)) mu(d) c_d(u)/phi(d). The summand is multiplicative in d, so the inner sum is prod_(p | base) (1 - c_p(u)/(p - 1)), and c_p(u) is p - 1 when p divides u and -1 otherwise, so the product is base/phi(base) when gcd(u, base) = 1 and 0 otherwise. At n >= 1, u is congruent to its last digit modulo base, so #{u in D_n : gcd(u, base) = 1} = fill^(n-1) #{f in F : gcd(f, base) = 1}. □

The rest of region C2 is Lambda less its mean on each class, and there Lambda differs from mu: a possible exceptional zero leaves a secondary term in the primes of a class, which Lemma 7.5 below bounds and Proposition 7.6 pays.

Lemma 7.3 (the constant on the half interval). At every odd base, kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill = base/(base + 1); at a prime base p it is p/(p + 1).

Proof. The residues prime to base pair as f and base - f, distinct because base is odd, and exactly one of each pair lies in {1, ..., (base-1)/2}, which is F less {0}; 0 is not prime to base. So #{f in F : gcd(f, base) = 1} = phi(base)/2 and kappa_F = base/(2 fill) = base/(base + 1). □

At a prime p this reads: (p-1)/2 of the (p+1)/2 last digits are prime to p, and each such class carries p/(p-1) times its share of the primes, so kappa_F = ((p-1)/(p+1)) (p/(p-1)) = p/(p+1).

The rest of this section proves part (ii). As in Sections 3 to 7 of the earlier paper, a block Py + D_n is fixed with x/y < base^K, where K = ceil(kappa sqrt(log x)) is the depth of its Proposition 3.3, kappa there a depth constant and not kappa_F; x is large, so log y >= (log x)/2; and every residue a mod y carries the fraction l/d of its Definition 4.1, with d <= Q, height h = |ad - ly| <= y/Q and |a/y - l/d| = h/(dy). The block sum of Lemma 7.2 is written Sigma_Lambda(P, n) = sum_(u in D_n) Lambda(Py + u), and S_P is as there.

Lemma 7.4 (the minor arcs for Lambda). For X >= 2 and real theta = l'/d' + beta with (l', d') = 1 and |beta| < 1/d'^2,

|sum_(v < X) Lambda(v) e(v theta)| << (X^(4/5) + X^(1/2) |d' beta|^(-1/2) + X |d' beta|^(1/2)) (log X)^4 ,

with an implied constant that the source does not print; its proof there involves no parameter, and this paper reads it as absolute. Consequently

|S_P(a/y)| << x^(4/5) base^(K/5) (log x)^4                          on region A ,
|S_P(a/y)| << (x^(4/5) base^(K/5) + x max(d, h)^(-1/2)) (log x)^4   on region B .

Proof. The first display is Lemma 4.2 of Maynard (2022), read at source with its proof in arXiv:1510.07711v1, which derives it from Vaughan's identity and its Lemma 4.1 on sum min(M, ||theta v||^(-1)). It says nothing at beta = 0, so it is read at a coarser fraction. Since d <= Q, h/y <= 1/Q <= 1/d. First, at d >= 2, Dirichlet's theorem at level floor(d/2) gives a reduced l'/d' with d' <= d/2 and |d' a/y - l'| < 2/d. Since d' < d and (l, d) = 1, d' l/d is not an integer, so |d' a/y - l'| >= |d' l/d - l'| - d' h/(dy) >= 1/d - 1/(2d). So beta' = a/y - l'/d' has 1/(2d) <= |d' beta'| < 2/d and |beta'| < 2/(d d') <= 1/d'^2, and the lemma gives (X^(4/5) + (2Xd)^(1/2) + X (2/d)^(1/2)) (log X)^4. Second, at l/d itself when h >= 1 and d < y^(2/5): there |beta| = h/(dy) < 1/d^2, since dh < y^(4/5), and |d beta| = h/y, so the lemma gives (X^(4/5) + (Xy/h)^(1/2) + X (h/y)^(1/2)) (log X)^4. S_P is the difference of at most two such sums with X <= 2x, and y <= x < y base^K. On region A, y^(2/5) <= d <= y^(3/5), and the first reading gives (2Xd)^(1/2) <= 2 x^(4/5) and X (2/d)^(1/2) <= 3 x y^(-1/5) < 3 x^(4/5) base^(K/5). On region B, d < y^(2/5) and max(d, h) >= Z. If d >= h, then d >= Z >= 2, and the first reading gives (2Xd)^(1/2) <= 2 x^(7/10) and X (2/d)^(1/2) <= 3 x d^(-1/2); if h > d, the second gives (Xy/h)^(1/2) <= 2 x h^(-1/2) and X (h/y)^(1/2) <= 2 x y^(-3/10) < 2 x^(7/10) base^(3K/10). In both cases every term but x max(d, h)^(-1/2) is at most a constant times x^(4/5) base^(K/5), since base^K < x. □

Lemma 7.4 takes the place of Lemma 4.3 of the earlier paper and has its shape. There the bound for mu holds at beta = 0 and is read at l/d and at a finer fraction; here it is read at a coarser fraction and at l/d, with (log x)^4 for (log x)^3 and no eps.

Lemma 7.5 (Lambda in progressions to base-smooth moduli). There are effective C_2, c_2' > 0, depending on base alone, such that for every X >= 2, every u <= X, every d dividing a power of base with d <= exp(sqrt(log X)/2) and every residue r,

|psi(u; d, r) - 1_(gcd(r,d)=1) u/phi(d)| <= C_2 X exp(-c_2' sqrt(log X)) ,

where psi(u; d, r) sums Lambda(v) over v <= u with v = r mod d.

Proof. Take X large; below any fixed X the bound holds by the choice of C_2. If u <= X exp(-sqrt(log X)), both terms are at most u log X, since psi(u) <= u log u. If gcd(r, d) > 1, every prime power counted shares a prime with d, so it is a power of a prime of base, and psi(u; d, r) <= omega(base) log u. Otherwise gcd(r, d) = 1, log u >= (log X)/2 and d <= u. At d >= 3, Theorem 12.4 of Koukoulopoulos (2019), read at source with its proof, gives

psi(u; d, r) = (u - chi_1(r) u^(beta_1)/beta_1)/phi(d) + O(u exp(-c_2 sqrt(log u)))

uniformly in u >= d >= 3 and gcd(r, d) = 1, with c_2 absolute, the second term present only when the L-functions modulo d have the one real exceptional zero beta_1 of its Theorem 12.3, of a real non-principal character chi_1. The display is in the form the proof of Theorem 12.4 produces, p. 122; the statement prints u^(beta_1) in place of u^(beta_1)/beta_1, and the two can differ by more than the error term, but both are at most 2 u^(1 - c_base) below, so either serves here. That proof uses the zero-free region, the explicit formula and a zero count, and the book's ineffectivity enters only through Siegel's theorem, which it does not use. beta_1 is a zero of the real primitive character inducing chi_1, whose conductor divides d and so divides 8 rad(base) by Lemma 6.4 of the earlier paper: one of a finite family fixed by the base. Theorem 12.8 of Koukoulopoulos (2019) puts every real zero of that family below 1 - c_base, with c_base > 0 effective and fixed by the base, so the secondary term is at most 2 u^(1 - c_base)/phi(d). At d <= 2 the same theorem at modulus 4 gives psi(u; d, r) = psi(u; 4, 1) + psi(u; 4, 3) + O(log u), the error counting the powers of 2, and the two secondary terms cancel, since chi_1(1) + chi_1(3) = 0. With log u >= (log X)/2, the error terms are at most a constant times X exp(-(c_2/2) sqrt(log X)) and 2 X exp(-c_base sqrt(log X)), so the lemma holds with c_2' = min(1/2, c_2/2, c_base). □

This is Lemma 6.5 of the earlier paper for Lambda. There the exceptional zero is met inside a Perron integral for 1/L; here it survives as a secondary term of the main term, and the same finite family of real characters, Lemma 6.4 there, bounds it. Siegel's theorem is never used, and this is the remark of Maynard (2022) that Siegel zeros play no role at these moduli, written out.

Proposition 7.6 (region C2 less its main term). Let rad(base) < Z <= min(2^n, exp(sqrt(log x)/2)), the second bound being C_0 <= 1/2 in Z = exp(C_0 sqrt(log x)), and rad(base) < Q, and let N_Z <= (1 + log_2 Z)^(omega(base)) be the number of d < Z dividing a power of base. Then for x large

|y^(-1) sum_(a in C2) hat F_n(a/y) S_P(-a/y) - kappa_F fill^n| <= 120 C_2 fill^n base^K Z^2 N_Z exp(-c_2' sqrt(log x)) .

Proof. By Lemma 7.2 the principal parts over C2 sum to kappa_F fill^n, so the left side is at most y^(-1) sum_(a in C2) |hat F_n(a/y)| |R(a)|, with R(a) the sum S_P(-a/y) less its principal part. At a residue of C2, a = ly/d + j with |j| = h/d < Z/d, as in the proof of Lemma 7.2, and e(-va/y) = e(-vl/d) e(-vj/y). Partial summation against e(-vj/y), whose total variation over the block is at most 2 pi |j|, gives |R(a)| <= (1 + 2 pi |j|) max_u |sum_(r mod d) e(-rl/d) E(u; d, r)| over Py <= u < (P+1) y, with the class error

E(u; d, r) = sum_(Py <= v <= u, v = r mod d) Lambda(v) - (d/phi(d)) 1_(gcd(r,d)=1) #{Py <= v <= u : v = r mod d} .

So |R(a)| <= (d + 2 pi Z) max |E| <= 8 Z max |E|. The count of a class up to a point w is within 1 of w/d, so |E(u; d, r)| is at most twice the left side of Lemma 7.5 plus 2 d/phi(d), and d/phi(d) <= base/phi(base). Lemma 7.5 at X = 2x applies, since u < 2x and d < Z <= exp(sqrt(log x)/2), and gives max |E| <= 4 C_2 x exp(-c_2' sqrt(log x)) + 2 base/phi(base) <= 5 C_2 x exp(-c_2' sqrt(log x)) for x large. C2 holds at most 3 Z N_Z residues, |hat F_n| <= fill^n on them, and x/y < base^K. □

This is Proposition 6.6 of the earlier paper with the class error of Lambda in place of M(u; d, b); the count of C2 and the partial summation are identical.

Proof of Theorem 7.1(ii). The block expansion and the split, Lemmas 3.1 and 3.2 of the earlier paper, hold with Lambda in place of mu, since neither reads the function beyond its being real. The reduction, Proposition 3.3 there, changes in two places. Its block hypothesis becomes |Sigma_Lambda(P, n) - kappa_F fill^n| <= C' fill^n exp(-c' sqrt(log x)) for every kept block, and the main terms kappa_F fill^n of the kept blocks differ from kappa_F A_F(x) by at most kappa_F times the number of elements of the discarded scales, the point x and the element 0. Since Lambda <= log x and kappa_F <= base/phi(base), the discarded scales weigh at most (log x + base/phi(base)) 6 fill^(1-K) A_F(x), and the point x and the element 0 at most log x + 2 kappa_F; the factor log x is absorbed by halving the exponent, and below any fixed x the bound (log x + kappa_F) A_F(x) is absorbed into C. So part (ii) follows from the block bound with c = min(c', kappa log fill)/2, as there.

Take

C_0 = min(1/2, sqrt(|log rho|/24), c_2'/4) ,   kappa = C_0/(20 log base) ,

the choice of the earlier paper with c_2' of Lemma 7.5 in place of its c_1, so that base^K <= base exp((C_0/20) sqrt(log x)), and rad(base) < Z <= 2^n and rad(base) < Q for x large. By the block expansion and Proposition 7.6, |Sigma_Lambda(P, n) - kappa_F fill^n| is at most the bound of Proposition 7.6 plus y^(-1) sum |hat F_n(a/y)| |S_P(a/y)| over each of A, B and C1, each fill^n times a saving.

• Region A: the certificate at s = 0 gives sum_(a mod y) |hat F_n(a/y)| <= C_F fill^n y^(alpha_1), and Lemma 7.4 with x^(4/5) < y^(4/5) base^(4K/5) gives << C_F fill^n y^(alpha_1 - 1/5) base^K (log x)^4, a power saving since log y >= (log x)/2. This is Proposition 4.4 of the earlier paper with no eps.
• Region B: Proposition 5.2 of the earlier paper with Lemma 7.4 in place of its Lemma 4.3. Its classes, D <= d < 2D and H <= h < 2H or h = 0, number at most (log_2 y + 2)^2, and on each max(d, h) >= max(D, H) > Z/2; its Lemma 5.1, under the certificate, bounds the mass of a class by << fill^n max(D, H)^(2 alpha_1), all as there. With Lemma 7.4 a class weighs << y^(-1) (x^(4/5) base^(K/5) + x max(D, H)^(-1/2)) (log x)^4 fill^n max(D, H)^(2 alpha_1). With max(D, H) <= y^(2/5) and x^(4/5) < y^(4/5) base^(4K/5) the first term is << fill^n base^K y^((4/5)(alpha_1 - 1/4)) (log x)^4; with x/y < base^K and 2 alpha_1 < 1/2 the second is << fill^n base^K (log x)^4 Z^(-(1/2 - 2 alpha_1)). Over the classes, region B is << fill^n base^K (log x)^6 (y^((4/5)(alpha_1 - 1/4)) + Z^(-(1/2 - 2 alpha_1))). It asks alpha_1 < 1/4, and 1/2 - 2 alpha_1 > 1/10 under a certificate below 1/5.
• Region C1: Proposition 6.3 of the earlier paper, through its Lemma 6.1 on the two consecutive digits, with the trivial bound |S_P| <= y log(2x) in place of |S_P| <= y, so the bound gains the factor log(2x): 3 rho^(-1) fill^n log(2x) exp((2 C_0 - |log rho|/(6 C_0)) sqrt(log x)).
• Region C2 less its main term: Proposition 7.6, a saving exp(-(c_2' - 2 C_0 - C_0/20) sqrt(log x)) up to the factor N_Z, a power of log x, and c_2' >= 4 C_0 makes that exponent at least (2 - 1/20) C_0.

As in the assembly of the earlier paper, every saving is at least exp(-(C_0/20) sqrt(log x)) up to powers of log x and constants depending on base and F, so the block bound holds with c' = C_0/21, and part (ii) follows with c = min(C_0/21, kappa log fill)/2. Every constant is effective: the certificate's C_F, the implied constant of Lemma 7.4, the constants c_2, c_base and C_2 of Lemma 7.5, and every threshold on x above. The bound c <= sqrt(|log rho|/24)/40 of the earlier paper holds for Lambda too. □

Proof of Theorem 1.1. At an odd base base >= 94939 the half interval contains the consecutive digits 0 and 1, and by Theorem 4.1 and Proposition 6.2 it carries the shifted-grid certificate C_F = 3/2, alpha_1 = log_base(lambda/fill) < 1/5. Theorem 7.1 gives both bounds, with kappa_F as stated; F is determined by base, so C and c depend on base alone. □

Proof of Corollary 1.2. By Lemma 2.2, at the prime base p the set S_F is {k >= 1 : p does not divide C(2k,k)}, so A(x) = A_F(x), the first sum is M_F(x), and the second is the von Mangoldt sum of Theorem 1.1, where kappa_F = p/(p+1) by Lemma 7.3. □

Proof of Theorem 1.3. Theorem 5.1 of The First Base Below a Quarter proves, under the stated hypothesis and for any digit set F with fill >= 2, dimension alpha_base and l^1 exponent alpha_1, that |M_F(x)| <<_(base, F, eps) A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha_base; its only Mobius input is the uniform bound max_theta |sum_(v <= x) mu(v) e(v theta)| << x^(3/4 + eps) of Baker and Harman (1991), and its alpha_1 is the growth exponent of fill^(-i) max_s Sigma_i(s). Theorem 4.1 bounds that exponent by log_base(lambda/fill), and a larger alpha_1 only weakens the conclusion, so the theorem holds with alpha_1 = log_base(lambda/fill), which is below 1/4 at every odd base from 3789 by Proposition 6.2. As the base grows, alpha_1 <= log((2/pi) log base + 2.64)/log base tends to 0 by Proposition 5.1 and alpha_base = log((base + 1)/2)/log base tends to 1, so delta tends to 1/4. □

What is in print

Every source below is in the references and read at source. The wider literature on missing-digit sets, with whole-text searches of the circle-method sources for a Mobius or Mertens sum, is Section 9 of the earlier paper, and none of the sources read there or here carries a Mobius or Mertens sum over a missing-digit set.

Maynard (2022), Theorem 1.3, read in arXiv:1510.07711v1, proves sum_(n < q^k) Lambda(n) 1_B(n) asymptotic with error O_A((q - s)^k (log q^k)^(-A)) when the s excluded digits are consecutive and q - s >= q^(4/5 + eps), q sufficiently large in terms of eps, by a sketch in its Section 9. The half interval has q - s = (q + 1)/2, inside that range, so the prime asymptotic on it at sufficiently large base, with an unquantified base and a log-power error, is the content of that theorem, and at a prime base so is the prime count on {k : q does not divide C(2k,k)}. The constant of the sketch, alpha_(q,s) = log((2 + 2/log q)(q/(q - s)) log q)/log q, clears 1/5 on the half interval only from q = 7777884825, the same reading giving the one-missing-digit crossing 1520573 of the constant of its Lemmas 5.1 and 5.3 (lab/py/interval-digits, verb wall); in the normalisation of Section 5 it pays 4 log base per digit where Theorem 4.1 pays (2/pi) log base, and the whole gain in the base lies there. The theorem prints its main term as q(phi(q) - s')/((q - 1) phi(q)) (q - s)^k, s' the excluded digits prime to q, which at the half interval of a prime q = p is p/(2(p - 1)) times the count, against the p/(p + 1) of Lemma 7.3. With q - s in place of q - 1, the form of the constant kappa_B = q(phi(q) - t)/(phi(q)(q - s)) of Maynard (2019), t the excluded digits prime to q, and of the instruction in Section 9 of Maynard (2022) to replace q - 1 by q - s, it is p/(p + 1); so the printed form reads as a misprint against the 2019 constant, and Lemma 7.2 gives the q - s form. After its Theorem 1.1, Maynard (2022) remarks that its estimates are used only at highly composite moduli, where Siegel zeros play no role, so its error terms could be replaced by effective ones of size O((q - 1)^k exp(-c k^(1/2))), and its Section 9 extends the arguments to Theorem 1.3 without restating the remark: for Lambda on the half interval the effective shape is read from that remark, not stated at source.

Maynard (2019), Theorem 1.2, read in arXiv:1604.01041v2, gives the order of magnitude X^(log(q - s)/log q)/log X of the primes avoiding the top block {q - s, ..., q - 1} at large q and s <= q - q^(57/80), a range that holds the half interval, and remarks the asymptotic for the primes avoiding the bottom block {0, ..., s - 1} at s <= q - q^(3/4 + delta), with an o(1) error. Neither paper carries a Mobius sum.

So for Lambda the asymptotic on the half interval is in print at sufficiently large base, and its effective shape is a reading of a remark made there at one missing digit; what this paper adds is the written proof of that shape, Section 7 from Lemma 7.2 on, the Mobius bound, which no source read here carries, and the explicit base 94939 for both sums.

The check and the limits

The certificate is a chain of elementary inequalities, and the links named at the end of Section 4 are recomputed on grids small enough to hold, where a wrong sign or a wrong count would show.

Fact 9.1 (the falsification). At every odd base 3..401 and at 1001, 10001 and 100001, on 201 shifts of [0, 1/2], the three sums of the proof of Theorem 4.1 stay under their bounds: the harmonic parts of Sigma_P and U, in units of base/pi, under log(base + 3) + gamma + K_2 t^2 and log(base + 4) - 0.0757, the first by at least 3.9999 * 10^(-5), and Sigma_Q under (base/pi) log tan(3 pi/8 + pi/(4 base)). At every odd base 3..401 on 801 shifts and at 1001, 4001, 10001, 30001 and 100001 on 401, the closed form of Lemma 3.3 meets the direct sum to relative 10^(-9), G stays under (4.2), at most 0.999857 of it and tight at t = 0 by design, T phi stays under lambda phi, at most 0.999866 of it, and G stays over the lower bound of Proposition 5.2, at least 1.534638 times it. The grid sums Sigma_i(s) at 40 digits, at the nine bases 3, 5, 7, 9, 11, 13, 15, 17, 21, the four shifts 0, 1/2, 1/(2 base) and (sqrt 5 - 1)/2, and every level with base^i <= 9261, from level 8 at base 3 to level 3 at base 21, stay under (3/2) lambda^i, at most 0.557408 of it, and the identity of Lemma 3.2 holds at level 2 at bases 5, 7 and 9 to relative 10^(-15). Every ratio is printed with directed rounding, upper bounds rounded up and lower bounds down (lab/py/interval-digits, verb check, 13 seconds).

The wall of Proposition 6.2 is where this certificate clears the bar, and the readings say the route itself stops not far below.

Conjecture 9.2 (the true rate). The growth rate rho_T of T on the half interval, the spectral radius that max_s Sigma_i(s) grows at, reads rho_T/fill = (2/pi) log base + 2.26 near base 7 * 10^4 and meets base^(1/5) between 70001 and 80001; the one-step constant reads max G/fill = (2 sqrt 2/pi) log base + 1.19, at t = 1/2, and clears 1/4 from 7075 and 1/5 from 317063.

Evidence. Power iteration of T with linear interpolation reads rho_T/fill - (2/pi) log base as 2.00766, 2.14698 and 2.21757 at 101, 1001 and 10001 on 1000 cells, and 2.25718, 2.26007 and 2.26252 at 60001, 70001 and 80001 on 300 cells, where base^(1/5) - rho_T/fill reads -0.2325, -0.0508 and 0.1137, and the iterated weight stays within a factor 0.81 of its maximum. The one-step reading G(1/2)/fill - (2 sqrt 2/pi) log base is 1.19055 at 7075 and 1.19171 at 317063, and the maximum of G over 401 shifts sits at t = 1/2 at every base checked from 1001 to 100001 (lab/py/interval-digits, verbs rate and check). Failure modes. The iteration interpolates and rounds, and bounds nothing: its low and high ratios agreeing to five digits says it converged on its grid, not that the grid resolves T. The constant beside (2/pi) log base still rises at 80001 and may not settle at 2.26, and the crossing is read at three bases. The one-step scan reads G at one point. If the conjecture holds, then since Sigma_i(0) = (T^i 1)(0) and the weight is bounded below, the unshifted mass of region A grows at rho_T too, so no shifted-grid certificate of any kind carries this route below the crossing.

Proposition 9.3 (uniform square-root cancellation implies RH). If for every eps > 0 some C_eps gives |M_F(x)| <= C_eps A_F(x)^(1/2 + eps) on the half interval at every odd base and every x >= 1, the Riemann hypothesis holds.

Proof. S_F contains every integer below fill, so at the odd base 2 ceil(x) + 1 the sum M_F(x) is the Mertens function M(x) and A_F(x) = floor(x). So M(x) = O(x^(1/2 + eps)) for every eps > 0, which is the Riemann hypothesis by Theorem 14.25 (C) of Titchmarsh (1986). □

The implication is trivial and the digits play no part in it: the range x < fill carries all of it, and there the statement is the Riemann hypothesis itself. No converse is claimed.

What the theorem does not say. The saving is exp(-c sqrt(log x)) with a tiny c, never a power, and C and c are computable but not printed. Below 94939 nothing is proved without a hypothesis, and below 3789 nothing under one. The theorem says nothing about the true size of M_F(x), for which square-root cancellation against A_F(x) is the natural guess, and Proposition 9.3 shows that guess, asked uniformly over the bases, is at least as strong as the Riemann hypothesis. It says nothing about the even-digit design 2 S_F of Remark 2.3.

Open problems

Three things are left undone. The even-digit design 2 S_F of Remark 2.3 has as its Mobius sum -sum mu(k) over the odd k of S_F up to x/2, and at an odd base the parity of k is that of its digit sum, so that sum is half of M_F(x/2) less its twist by e(k/2); the twist reads the certificate at the shift 1/2, which Theorem 4.1 covers, but near a fraction l/d with d dividing a power of the base it sees the transform at l/d + 1/2, so regions C1 and C2 must be rerun with 2 base in place of base, and that is not written. No proof is written below 94939: a sharper weight than 1 + |sin(pi t)|/2 moves the wall toward the crossing of the true rate, which Conjecture 9.2 puts between 70001 and 80001, and if the readings are right no certificate passes that crossing, leaving the exponent 4/5 of the minor-arc input as the other lever, which neither this paper nor Maynard (2022) moves. And the growth rate of T itself is read, not proved: the constant beside (2/pi) log base in rho_T/fill lies between -1.31 and 2.60043004 + 3.4/base by Section 5 and reads near 2.26, and whether it settles is open.

Reproducibility

One study prints every number of Sections 4, 5, 6, 8 and 9, run from the repository root with one verb and raising if any check fails. uv run python research/lab/py/interval-digits/interval.py wall, under a second, prints Propositions 5.1, 5.2 and 6.2, the constant c_inf and the crossing 7777884825 of the Section 9 constant of Maynard (2022) with its calibration 1520573; check, in about 13 seconds, prints Fact 9.1 and the position of the maximum of G; rate, in about 20 seconds, prints the readings of Conjecture 9.2. The study evaluates every closed form at 120 bits in mpmath interval arithmetic, lower bounds truncated down and upper bounds rounded up; check runs in float64 with margins far above float error and prints its ratios with directed rounding, except at t = 0, where (4.2) is tight to 1.5 * 10^(-4) of its size by design, and its level sums at 40 digits; rate prints readings that bound nothing. The figure is bash scripts/figures.sh paper-half-interval-mobius, under half a second a theme; its binary recomputes the closed form (4.1) in floating point at about 1600 odd bases from 101 to 10^7 + 1 and at the four bases beside the walls, and asserts the walls 94939 and 3789, the sign of both margins at every base it draws, the margins 1.3678 * 10^(-8) and 7.9625 * 10^(-6) to five digits and alpha_1 at 100003. Theorem 7.1 prints no number. Part (i) is reused, and the numbers of its dissection are printed by the studies named in the earlier paper. The proof of part (ii) computes no constant; its main-term identity and its four regions for Lambda are checked on small designs far below any wall by uv run python research/lab/py/prime-dissection/primes.py series and regions, each in about a second, a check of the bookkeeping and not of the theorem.

References

• Maynard 2022, Primes and polynomials with restricted digits, Int. Math. Res. Not. 2022, 10626-10648, read in arXiv:1510.07711v1. doi.org/10.1093/imrn/rnab002
• Maynard 2019, Primes with restricted digits, Invent. Math. 217, 127-218, read in arXiv:1604.01041v2. link.springer.com
• Basak, Robles and Zaharescu 2023, Exponential sums over Mobius convolutions with applications to partitions. arxiv.org/abs/2312.17435
• Koukoulopoulos 2019, The Distribution of Prime Numbers, Graduate Studies in Mathematics 203, American Mathematical Society, doi:10.1090/gsm/203, read in the author's preliminary version. dms.umontreal.ca
• Chang and Martin 2019, The smallest invariant factor of the multiplicative group. arxiv.org/abs/1908.00035
• Baker and Harman 1991, Exponential sums formed with the Mobius function, J. London Math. Soc. (2) 43, 193-198. doi.org/10.1112/jlms/s2-43.2.193
• Titchmarsh 1986, The Theory of the Riemann Zeta-Function, second edition revised by D. R. Heath-Brown, Clarendon Press, Oxford, ISBN 978-0-19-853369-6, Theorem 14.25 (C); the link is the publisher's record. global.oup.com
• Kummer 1852, Uber die Erganzungssatze zu den allgemeinen Reciprocitatsgesetzen, J. reine angew. Math. 44, 93-146, doi:10.1515/crll.1852.44.93; the link is the record with the public-domain scan. eudml.org/doc/147500