franel-converse.md

Take the reduced fractions a/b in (0, 1] with b <= Q and b written, in base q, with digits from a fixed set F of at least two digits, and record how far the j-th of them sits from j/m, m their number. On the full set Franel proved in 1924 that the sum of the squares of those distances is O(Q^(-1+eps)) for every eps > 0 exactly when the Riemann hypothesis holds, through the Mertens function M(x). On a design the identity behind that theorem splits: its Fourier side is a gcd-weighted quadratic form in one Mertens sum for every dilate {c : dc in S_F} of the design, sets that collapse to the whole line only when every digit is allowed. Jordan's totient writes the form as a sum of squares, Schur's test bounds the change of variables by (1 + ln Q)^2 both ways, and the threshold sum_j delta_j^2 = O(Q^(-1+eps)) becomes equivalent to sum_(d <= Q) M_F(Q/d; d)^2 = O(Q^(alpha+eps)), alpha = log |F| / log q; a primorial witness shows that no constant replaces the logarithms. The masses of the dilates sum to O(Q^(alpha+eps)) with no Mobius input, so square-root cancellation of mu in each dilate against its own mass gives the threshold on every design; a hypothesis with an explicit rate in d also suffices and is not shown to be needed, and the equidistribution law behind its rate fails at the repunits of base 3 {0,1}. Every identity is checked as an exact rational at base 3 {0,1} to Q = 3^6, in floating point to Q = 3^8, and on the full set at Q = 40, 81, 243.

Introduction

Write every whole number in base 3 and keep the ones that use only the digits 0 and 1: 1, 3, 4, 9, 10, 12, 13, 27, ..., sixteen of them up to 81. Those are the denominators. The fractions a/b with such a b, 1 <= a <= b and gcd(a, b) = 1, sorted, are the top row of the figure, 241 of them at Q = 81; the middle row is the full Farey sequence of order 81, 2020 fractions. Number the design's fractions rho_1 < ... < rho_m and set delta_j = rho_j - j/m; the orange sawtooth of the bottom row is delta_j against rho_j, the full set's sawtooth behind it in grey on the same scale. Both wander a little above and below zero and both come back. How fast they come back, in the square mean, is the question.

For the full set the answer is Franel's theorem (1924): sum_j delta_j^2 = O(Q^(-1+eps)) for every eps > 0 if and only if the Riemann hypothesis holds; Landau (1924), in the note printed after Franel's, gave the same for sum_j |delta_j| = O(Q^(1/2+eps)), and Edwards reproduces both in section 12.2 through the Mertens function M(x) = sum_(n <= x) mu(n). The mechanism is an identity: the sum of squares is, up to a constant and a factor 12 m, the form sum_(d, e) (gcd(d, e)^2/(d e)) M(Q/d) M(Q/e), which is O(Q^(1+eps)) under M(x) = O(x^(1/2+eps)) and bounds M(Q)^2 from above. On a denominator restricted to a digit design the Farey page of this tree finds, and this paper proves, that the two directions no longer meet at one function: the form becomes sum_(d, e) (gcd(d, e)^2/(d e)) M_F(Q/d; d) M_F(Q/e; e) with M_F(x; d) = sum_(c <= x, dc in S_F) mu(c) the Mertens sum of the dilate {c : dc in S_F}, a different set for every d. The d = 1 term is the design's own M_F(Q), so the threshold still forces M_F(Q) = O(Q^(alpha/2+eps)); the converse needs every dilate.

The theorem of this paper is that the converse needs them in mean square and in nothing finer: for every digit design with at least two digits, sum_j delta_j^2 = O(Q^(-1+eps)) on the denominator-restricted set if and only if sum_(d <= Q) M_F(Q/d; d)^2 = O(Q^(alpha+eps)), the form lying between (6/pi^2)(1 + ln Q)^(-2) and (1 + ln Q)^2 times the mean square for every design and every Q >= 1 (Theorem 4.3), with no constant in place of the logarithms (Proposition 4.4). On the full set every dilate is the whole line and the statement is Franel's with M(x) = O(x^(1/2+eps)) standing for the hypothesis (Corollary 4.6). Section 2 sets up the objects; Section 3 proves the three identities; Section 4 the sandwich and the equivalence; Section 5 shows that the masses of the dilates sum to O(Q^(alpha+eps)) with no Mobius input, so that square-root cancellation of mu in each dilate against its own mass gives the threshold, and that a hypothesis with an explicit rate in d suffices as well; Section 6 proves that the equidistribution law behind that rate fails at the repunits; Section 7 prints the readings; Section 8 says what is open.

The literature is positioned as read here, four texts: Franel's identity is stated over the rationals in its Fourier form by Huxley (1971, Lemma (10)) and with weights by Huxley (2012, Theorem 1), extended by Kanemitsu and Yoshimoto (1996, Theorem 3) and surveyed by Cobeli and Zaharescu (2003); none restricts the denominators to a digit design, the older literature on Farey fractions with restricted denominators is not surveyed, and this paper claims the theorems it proves and no priority beyond them. The divisor form of Ramanujan's sum is equation (2.7) of Ramanujan (1918); the factorization of a gcd matrix through a totient is Smith's (1875), and Lemma 4.1 is it with Jordan's totient; the operator bound is Schur's (1911); the equivalence of M(x) = O(x^(1/2+eps)) with the Riemann hypothesis is Theorem 14.25 (C) of Titchmarsh (1986).

Definitions

Definition 2.1 (design). Fix a base q >= 2 and a set F of digits, F a subset of {0, ..., q-1} with |F| >= 2. The digit design S_F is the set of whole numbers whose every base-q digit lies in F, with 0 in S_F. Its dimension is alpha = log |F| / log q, in (0, 1], and A_F(Q) = #{n in S_F : 1 <= n <= Q}. The full set is F = {0, ..., q-1}, where S_F is every whole number and alpha = 1.

Lemma 2.2. For Q >= q, Q^alpha / |F|^2 < A_F(Q) < 2 |F| Q^alpha.

Proof. Let f = |F| and L = floor(log_q Q), so q^L <= Q < q^(L+1) and L >= 1. Every member of S_F in [1, Q] has between 1 and L + 1 digits, all in F, and there are at most f^k such strings of length k, so A_F(Q) <= sum_(k = 1)^(L+1) f^k = (f^(L+2) - f)/(f - 1) < (f^2/(f - 1)) f^L <= 2 f q^(L alpha) <= 2 f Q^alpha, using f/(f - 1) <= 2. Some digit of F is nonzero, so the strings of exactly L digits with that leading digit and the other L - 1 digits free in F are at least f^(L-1) members of S_F, each in [q^(L-1), q^L), hence at most Q; and f^(L-1) = q^((L+1) alpha) / f^2 > Q^alpha / f^2. □

The upper bound holds with |F| in place of 2|F| when 0 in F, where every member is a padded string of L + 1 digits, and not otherwise: at base 4 with F = {1, 2} and Q = 41, A_F(41) = 13 against |F| Q^alpha = 12.81.

Definition 2.3 (denominator set and sawtooth). For Q >= 1 the denominator-restricted Farey set is F_Q(S_F) = {a/b : b in S_F, 1 <= b <= Q, 1 <= a <= b, gcd(a, b) = 1}, of size m_F(Q) = sum_(b in S_F, b <= Q) phi(b) with phi Euler's totient. Its nodes ascending are rho_1 < ... < rho_m, m = m_F(Q), and delta_j = rho_j - j/m. Its exponential sum at the integer frequency k is S_F(k, Q) = sum_(j) e(k rho_j) with e(x) = exp(2 pi i x).

Lemma 2.4. phi(n) > n / (e^2 (ln n)^2) for n >= 16, and for Q >= 16 q^2, Q^(1+alpha) / (e^2 q^2 |F|^2 (ln Q)^2) < m_F(Q) < 2 |F| Q^(1+alpha).

Proof. Put y = (ln n)^2 >= 2. Over the primes p <= y dividing n, p/(p-1) <= exp(1/(p-1)) and the values p - 1 are distinct integers in [1, y], so their product is at most exp(sum_(k <= y) 1/k) <= exp(1 + ln y) = e y. The primes p > y dividing n number at most ln n / ln y, since their product is at most n, and each contributes p/(p-1) <= exp(1/(p-1)) <= exp(2/y), so together at most exp(2 ln n / (y ln y)) = exp(1/(ln n ln ln n)) < e. Hence n/phi(n) = prod_(p | n) p/(p-1) < e^2 (ln n)^2. For the node count, m_F(Q) <= sum_(b in S_F, b <= Q) b <= Q A_F(Q) < 2 |F| Q^(1+alpha) by Lemma 2.2; and the |F|^(L-1) > Q^alpha / |F|^2 members of exactly L = floor(log_q Q) digits found there each exceed q^(L-1) >= Q/q^2 >= 16, so each has phi(b) > (Q/q^2)/(e^2 (ln Q)^2). □

Definition 2.5 (dilated Mertens sums). For d >= 1 the dilate of the design is d^(-1) S_F = {c >= 1 : dc in S_F}, its counting function N_F(Q; d) = #{c : 1 <= c <= Q/d, dc in S_F} = #{n in S_F : 1 <= n <= Q, d | n}, and its Mertens sum M_F(x; d) = sum_(1 <= c <= x, dc in S_F) mu(c), with mu the Mobius function. At d = 1 it is the design's own Mertens sum M_F(x) = sum_(n in S_F, n <= x) mu(n) of the Mobius page. Throughout, x_d = M_F(Q/d; d) for 1 <= d <= Q, a vector of integers with |x_d| <= N_F(Q; d) termwise, and x_d = 0 for d > Q. On the full set every dilate is the whole line and x_d = M(floor(Q/d)).

For a design carrying 0 a dilate is a regular language, and the Farey page proves it: long multiplication by d, read from the least significant digit, is a deterministic automaton on the d carries, from carry r the digit e writing the output digit (de + r) mod q, which must lie in F, and moving to the carry floor((de + r)/q), with acceptance when the terminal carry lies in Acc_d = {0} union (S_F intersect [1, d)); every column of its transfer matrix sums to |F|, so a dilate carries the design's own dimension. Nothing below uses the automaton beyond one consequence, which is elementary:

Lemma 2.6 (base powers are free). If 0 in F then qc in S_F if and only if c in S_F, so M_F(x; q^j d) = M_F(x; d) and N_F(q^j Q; q^j d) = N_F(Q; d) for every j >= 0.

Proof. Appending or removing a zero digit neither enters nor leaves S_F. □

The identities

Lemma 3.1 (the divisor form of Ramanujan's sum). For b, m >= 1, c_b(m) = sum_(a mod b, gcd(a, b) = 1) e(ma/b) = sum_(d | gcd(m, b)) d mu(b/d).

Proof. Insert sum_(g | gcd(a, b)) mu(g) = [gcd(a, b) = 1] and write a = g a': c_b(m) = sum_(g | b) mu(g) sum_(a' mod b/g) e(m a' / (b/g)). The inner sum is b/g when (b/g) | m and 0 otherwise; substitute d = b/g. □

This is equation (2.7) of Ramanujan (1918), credited there to Landau's Handbuch, page 577; the proof is included so that nothing rests on the citation.

Theorem 3.2 (the exponential sum at frequency m). For every m >= 1 and Q >= 1, S_F(m, Q) = sum_(d | m) d M_F(Q/d; d). In particular S_F(1, Q) = M_F(Q).

Proof. Group the nodes by denominator: S_F(m, Q) = sum_(b in S_F, b <= Q) c_b(m), and c_b(m) depends on b alone. By Lemma 3.1 this is sum_(b in S_F, b <= Q) sum_(d | m, d | b) d mu(b/d) = sum_(d | m) d sum_(c <= Q/d, dc in S_F) mu(c), writing b = dc. □

Theorem 3.3 (the Fourier form). With x_d = M_F(Q/d; d) and the form G_F(Q) = sum_(d, e >= 1) (gcd(d, e)^2/(d e)) x_d x_e, a finite sum of rationals, sum_(k != 0) |S_F(k, Q)|^2 / k^2 = (pi^2/3) G_F(Q).

Proof. The node set is carried to itself by r -> 1 - r away from the node 1, which e(k r) sends to 1, so S_F(k, Q) is real and S_F(-k, Q) = S_F(k, Q). By Theorem 3.2, for k >= 1, S_F(k, Q)^2 = sum_(d | k) sum_(e | k) d e x_d x_e, and summing k^(-2) over the k >= 1 divisible by both d and e, that is by lcm(d, e), gives zeta(2)/lcm(d, e)^2; the exchange is over finitely many d, e <= Q and an absolutely convergent series in k. So sum_(k != 0) |S_F(k, Q)|^2/k^2 = 2 zeta(2) sum_(d, e) (d e/lcm(d, e)^2) x_d x_e, and d e / lcm(d, e)^2 = gcd(d, e)^2/(d e) with 2 zeta(2) = pi^2/3. □

Theorem 3.4 (the rank form). Let rho_1 < ... < rho_m be any m >= 1 points of (0, 1], S(k) = sum_j e(k rho_j), delta_j = rho_j - j/m and s = sum_j rho_j - m/2. Then sum_(k != 0) |S(k)|^2 / k^2 = 4 pi^2 (m sum_j delta_j^2 + s(1 - s) - 1/6).

Proof. Let B(x) = {x} - 1/2 be the sawtooth, with the Fourier series B(x) = -sum_(k != 0) e(kx)/(2 pi i k) in L^2. For u in [0, 1), {u + rho_j} = u + rho_j - [rho_j >= 1 - u], so with A(v) = #{j : rho_j < v} and D(v) = A(v) - m v, sum_j B(u + rho_j) = m u + sum_j rho_j - m/2 - (m - A(1 - u)) = D(1 - u) + s. Parseval on [0, 1] reads sum_(k != 0) |S(k)|^2/(4 pi^2 k^2) = int_0^1 (D(v) + s)^2 dv = int_0^1 D^2 - s^2, because int_0^1 A(v) dv = sum_j (1 - rho_j) gives int_0^1 D = -s. Put rho_0 = 0, rho_(m+1) = 1 and u_j = m rho_j - j, so u_0 = 0 and u_(m+1) = -1. On (rho_j, rho_(j+1)] the function A is j, and int_(rho_j)^(rho_(j+1)) (j - mv)^2 dv = ((j - m rho_j)^3 - (j - m rho_(j+1))^3)/(3m) = (-u_j^3 + (u_(j+1) + 1)^3)/(3m). Summing over j = 0, ..., m, the term (u_(m+1) + 1)^3 vanishes, u_0^3 vanishes, and what is left is (1/(3m)) sum_(j = 1)^m (3 u_j^2 + 3 u_j + 1) = m sum_j delta_j^2 + sum_j delta_j + 1/3, using u_j = m delta_j. Finally sum_j delta_j = sum_j rho_j - (m+1)/2 = s - 1/2, so int_0^1 D^2 - s^2 = m sum_j delta_j^2 + s(1 - s) - 1/6. □

Corollary 3.5. On the denominator set with m_F(Q) >= 1, G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + 1 when 1 in F and G_F(Q) = 12 m_F(Q) sum_j delta_j^2 - 2 when 1 is not in F. In either case G_F(Q) >= 0 and G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + O(1).

Proof. The node 1 = 1/1 is present exactly when 1 in S_F, that is when 1 in F; for b >= 2 the nodes a/b with a < b pair under a -> b - a and sum to phi(b)/2. So s = 1/2 when 1 in F and s = 0 otherwise, and Theorems 3.3 and 3.4 give (pi^2/3) G_F(Q) = 4 pi^2 (m sum_j delta_j^2 + 1/12) or 4 pi^2 (m sum_j delta_j^2 - 1/6). The left side of Theorem 3.3 is a sum of nonnegative terms. □

The constant is read as an exact rational at every Q <= 40 at which the node set changes: 1 on base 3 {0,1} and base 10 without the digit 9, -2 on base 3 {0,2}, base 4 {0,2,3}, base 5 {0,2,4} and base 10 {0,2,5,7} (lab/py/restricted-franel, verb sandwich). On the full set the identity is the one Edwards proves in section 12.2.

The sandwich and the equivalence

Lemma 4.1 (the Jordan form). Let J_2(f) = f^2 prod_(p | f) (1 - p^(-2)) be Jordan's totient and, for f <= Q, y_f = sum_(m <= Q/f) x_(fm)/m. Then G_F(Q) = sum_(f <= Q) (J_2(f)/f^2) y_f^2, with every weight J_2(f)/f^2 in [6/pi^2, 1]. Hence (6/pi^2) sum_f y_f^2 <= G_F(Q) <= sum_f y_f^2 for every real vector x.

Proof. J_2 = mu * id^2 as Dirichlet convolution, so sum_(f | n) J_2(f) = n^2 by Mobius inversion, and gcd(d, e)^2 = sum_(f | d, f | e) J_2(f). Then G_F(Q) = sum_f J_2(f) (sum_(f | d) x_d/d)^2 = sum_f (J_2(f)/f^2) (sum_m x_(fm)/m)^2, and prod_(p | f) (1 - p^(-2)) lies between prod_p (1 - p^(-2)) = 1/zeta(2) and 1. □

The vector y is itself a family of dilated sums: writing n = mc, y_f = sum_(n <= Q/f, fn in S_F) w(n) with w = mu * (1/id), |w(n)| = phi(rad n)/n <= 1.

Lemma 4.2 (Schur's test). For a real matrix A = (a_(ij)) with largest absolute row sum R and largest absolute column sum C, ||A x||^2 <= R C ||x||^2 in l^2.

Proof. By Cauchy and Schwarz, (sum_j a_(ij) x_j)^2 <= (sum_j |a_(ij)|)(sum_j |a_(ij)| x_j^2) <= R sum_j |a_(ij)| x_j^2; summing over i and exchanging, ||A x||^2 <= R sum_j x_j^2 sum_i |a_(ij)| <= R C ||x||^2. □

Theorem 4.3 (the sandwich). Let H_Q = sum_(k <= Q) 1/k and N_Q = H_Q max_(d <= Q) sigma(d)/d, with sigma the sum of divisors. Then for every real vector x indexed by d <= Q, sum_f y_f^2 <= N_Q sum_d x_d^2 and sum_d x_d^2 <= N_Q sum_f y_f^2, and N_Q <= H_Q^2 <= (1 + ln Q)^2. Consequently, for every digit design and every Q >= 1,

(6/pi^2) (1 + ln Q)^(-2) sum_(d <= Q) M_F(Q/d; d)^2  <=  G_F(Q)  <=  (1 + ln Q)^2 sum_(d <= Q) M_F(Q/d; d)^2 .

Proof. The map x -> y is the upper triangular matrix with entry 1/m at (f, fm), and it is inverted by the matrix with entry mu(m)/m at (f, fm): sum_m (mu(m)/m) sum_n x_(fmn)/n = sum_k (x_(fk)/k) sum_(m | k) mu(m) = x_f. Row f of either matrix has absolute sum sum_(m <= Q/f) 1/m <= H_Q. Column d of the first has absolute sum sum_(f | d) f/d = sigma(d)/d; column d of the inverse has absolute sum sum_(m | d) |mu(m)|/m = prod_(p | d) (1 + 1/p) <= sigma(d)/d. And sigma(d)/d = sum_(k | d) 1/k <= H_d <= H_Q. Lemma 4.2 bounds both squared operator norms by N_Q <= H_Q^2, and H_Q <= 1 + ln Q. The display combines the two norm bounds with Lemma 4.1. □

Robin's unconditional bound, sigma(d)/d < e^gamma ln ln d + 0.6483/ln ln d for d >= 3, quoted here from the Farey page as Theorem 2 of Robin (1984) and not read at source, sharpens N_Q to O(ln Q ln ln Q).

Proposition 4.4 (no constant). The ratio sum_f y_f^2 / sum_d x_d^2 is unbounded over the real vectors x indexed by d <= Q as Q grows: for x_d = 1 at the divisors of a primorial N = prod_(p <= z) p <= Q and 0 elsewhere, the ratio is prod_(p <= z) (1 + 1/p + 1/(2 p^2)) > sum_(p <= z) 1/p.

Proof. For f | N, y_f = sum_(m | N/f) 1/m = sigma(N/f)/(N/f), and y_f = 0 otherwise. So sum_f y_f^2 = sum_(g | N) (sigma(g)/g)^2 = prod_(p <= z) (1 + (1 + 1/p)^2) by multiplicativity over the squarefree N, while sum_d x_d^2 = 2^(pi(z)); the ratio is prod_(p <= z) (1 + 1/p + 1/(2p^2)), which exceeds 1 + sum_(p <= z) 1/p. By the elementary bound prod_(p <= z) p < 4^z, the primorial of z = ln Q / ln 4 is at most Q, and sum_(p <= z) 1/p diverges with z. □

By Mertens' theorem on the prime product (1874) the ratio is of order ln z, so with z of order ln Q it grows like ln ln Q; the witness rules out a constant and says nothing about whether N_Q is attained.

Theorem 4.5 (the mean-square Franel equivalence). For every digit design with at least two digits the following are equivalent, each read for every eps > 0: (i) sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted set; (ii) G_F(Q) = O_eps(Q^(alpha+eps)); (iii) sum_(f <= Q) y_f^2 = O_eps(Q^(alpha+eps)); (iv) sum_(d <= Q) M_F(Q/d; d)^2 = O_eps(Q^(alpha+eps)).

Proof. For Q >= 16 q^2 the node count is at least 1, and by Corollary 3.5 G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + kappa with kappa in {1, -2}, while Lemma 2.4 gives Q^(1+alpha)/(C (ln Q)^2) < m_F(Q) < 2 |F| Q^(1+alpha). Under (i), G_F(Q) <= 24 |F| Q^(1+alpha) C_eps Q^(-1+eps) + 1, which is (ii); under (ii), sum_j delta_j^2 = (G_F(Q) - kappa)/(12 m_F(Q)) <= (G_F(Q) + 2)/(12 m_F(Q)) << Q^(alpha+eps) (ln Q)^2 Q^(-1-alpha), which is (i) with 2 eps. Lemma 4.1 gives (ii) if and only if (iii) with constants 6/pi^2 and 1. Theorem 4.3 gives (iii) if and only if (iv) at the cost of (1 + ln Q)^2, absorbed by the eps. □

Corollary 4.6 (the full set is Franel). On the full set the four statements of Theorem 4.5 are each equivalent to M(x) = O_eps(x^(1/2+eps)), hence to the Riemann hypothesis.

Proof. Every dilate is the whole line, so (iv) reads sum_(d <= Q) M(Q/d)^2 = O(Q^(1+eps)). Its d = 1 term is M(Q)^2, so (iv) gives M(Q) = O(Q^(1/2+eps)). Conversely |M(x)| <= C_eps x^(1/2+eps) gives sum_(d <= Q) M(Q/d)^2 <= C_eps^2 Q^(1+2eps) sum_(d >= 1) d^(-1-2eps) = C_eps^2 zeta(1+2eps) Q^(1+2eps). That M(x) = O_eps(x^(1/2+eps)) for every eps is equivalent to the Riemann hypothesis is Theorem 14.25 (C) of Titchmarsh (1986). □

Landau's criterion is recovered one way only: by Cauchy and Schwarz sum_j |delta_j| <= (m sum_j delta_j^2)^(1/2), so (i) gives sum_j |delta_j| = O(Q^((alpha+eps)/2)), Landau's O(Q^(1/2+eps)) on the full set; the converse is Landau's own argument and is not derived here.

Corollary 4.7 (the ceiling). On every digit design, (i) implies M_F(Q) = O_eps(Q^(alpha/2+eps)), since M_F(Q)^2 = x_1^2 <= sum_d x_d^2.

The surrogate and one hypothesis

Lemma 5.1 (the divisor bound). For every eps > 0 there is C_eps with tau(m) <= C_eps m^eps for all m >= 1, tau the number of divisors.

Proof. tau(m) m^(-eps) = prod (k + 1) p^(-k eps) over the prime powers p^k exactly dividing m. A factor with p >= 2^(1/eps) is at most (k + 1) 2^(-k) <= 1; a factor with p < 2^(1/eps) is at most c_eps = max_(k >= 0) (k + 1) 2^(-k eps), and there are fewer than 2^(1/eps) such primes. So C_eps = c_eps^(2^(1/eps)) serves. □

Theorem 5.2 (the surrogate is a theorem). Let B(Q) = sum_(d, e <= Q) (gcd(d, e)^2/(d e)) sqrt(N_F(Q; d) N_F(Q; e)). For every digit design and every Q >= 1,

sum_(n in S_F, n <= Q) tau(n)  <=  B(Q)  <=  (1 + ln Q) sum_(d <= Q) (sigma(d)/d) N_F(Q; d)  <=  (1 + ln Q)^2 sum_(n in S_F, n <= Q) tau(n) ,

hence A_F(Q) <= B(Q) <= C_eps (1 + ln Q)^2 Q^eps A_F(Q) and B(Q) = O_eps(Q^(alpha+eps)), with no Mobius function anywhere.

Proof. Every term of B(Q) is nonnegative and the diagonal d = e has kernel 1, so B(Q) >= sum_(d <= Q) N_F(Q; d) = sum_(n in S_F, n <= Q) tau(n), exchanging the count of pairs (d, n) with d | n. For the upper bound, sqrt(N_F(Q; d) N_F(Q; e)) <= (N_F(Q; d) + N_F(Q; e))/2 and the kernel is symmetric, so B(Q) <= sum_(d <= Q) N_F(Q; d) sum_(e <= Q) gcd(d, e)^2/(d e). Group the inner sum by f = gcd(d, e), a divisor of d, and write e = f e': the kernel is f/(d e') and the sum over e' <= Q/f is at most H_Q, so the inner sum is at most (1/d) sum_(f | d) f H_Q = (sigma(d)/d)(1 + ln Q). Exchanging again, sum_(d <= Q) (sigma(d)/d) N_F(Q; d) = sum_(n in S_F, n <= Q) sum_(d | n) sigma(d)/d, and since sigma(d)/d = sum_(k | d) 1/k grows along divisibility, sum_(d | n) sigma(d)/d <= tau(n) sigma(n)/n <= tau(n)(1 + ln n). Lemma 5.1 and Lemma 2.2 finish. □

Theorem 5.3 (one hypothesis). Call (SR) the statement that for every eps > 0 there is C_eps with |M_F(Q/d; d)| <= C_eps N_F(Q; d)^(1/2 + eps) for every 1 <= d <= Q and every Q >= 1: square-root cancellation of mu in each dilate against that dilate's own mass. Under (SR), sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted set of every digit design with at least two digits.

Proof. Square (SR) and sum: N_F(Q; d)^(1+2eps) <= A_F(Q)^(2eps) N_F(Q; d), so sum_(d <= Q) M_F(Q/d; d)^2 <= C_eps^2 A_F(Q)^(2eps) sum_(n in S_F, n <= Q) tau(n) = O(Q^(alpha + eps(1 + 2 alpha))) by Theorem 5.2 and Lemma 2.2, and Theorem 4.5 turns the mean square into the threshold. Through the kernel the same line reads G_F(Q) <= C_eps^2 A_F(Q)^(2eps) B(Q). □

So the converse rests on cancellation of mu alone, pointwise in each dilate against its own mass, and on no statement about how S_F sits in residue classes; at d = 1 (SR) is the square-root conjecture for the design's Mertens sum, at d = q^j the same by Lemma 2.6, at every other d new. The threshold is not shown to return (SR): the one route tried, the termwise bound |S_F(k, Q)| <= k (pi^2 G_F(Q)/6)^(1/2) from Theorem 3.3 with the Mobius inversion d M_F(Q/d; d) = sum_(c | d) mu(d/c) S_F(c, Q) of Theorem 3.2, yields only |M_F(Q/d; d)| <= (sigma(d)/d)(pi^2 G_F(Q)/6)^(1/2), which grows in d where (SR) asks for the size of the dilate; no counterexample to the implication is known, and none can be exhibited while both sides are open.

Proposition 5.4 (the hypothesis with a rate). Write d_co for the part of d coprime to q and call (U') the bound |M_F(x; d)| <= C_eps d_co^((alpha-1)/2) x^(alpha/2 + eps) uniform in d >= 1 and x >= 1. Then (U') implies (iv) of Theorem 4.5, hence the threshold; and the bound (U) with d in place of d_co holds for no design carrying both 0 and 1 with alpha < 1.

Proof. Write d = a d_co with every prime of a dividing q. Under (U'), sum_(d <= Q) M_F(Q/d; d)^2 <= C_eps^2 Q^(alpha + 2eps) sum_d d_co^(alpha-1) d^(-alpha-2eps) = C_eps^2 Q^(alpha+2eps) sum_(a, d_co) a^(-alpha-2eps) d_co^(-1-2eps), and the double sum is at most prod_(p | q) (1 - p^(-alpha-2eps))^(-1) zeta(1 + 2eps), finite because alpha > 0. For (U), Lemma 2.6 gives M_F(x; q^j) = M_F(x), so (U) at d = q^j would read |M_F(x)| <= C_eps q^(j(alpha-1)/2) x^(alpha/2+eps) for every j, and since alpha < 1 the right side tends to 0 at fixed x, forcing M_F(x) = 0 against M_F(1) = mu(1) = 1. □

(U') asks, at x = Q/d, that |M_F(Q/d; d)| be at most (Q^alpha/d_co)^(1/2) up to Q^(O(eps)) and the base-smooth factor: it is (SR) with the mass N_F(Q; d) replaced by the main term A_F(Q)/d that equidistribution of S_F in the residue class 0 mod d would give. The next section shows that the main term is not a uniform upper bound for the mass, so (SR) and (U') are different hypotheses: at the repunits of Theorem 6.1 the mass exceeds A_F(Q)/d by a factor growing like (3/2)^t, and (U') asks there for more cancellation than (SR) does.

What fails: the uniform law at the repunits

The accepting-set law is the statement N_F(Q; d) ~ A_F(Q)/d at fixed d coprime to the base, the design equidistributed in the residue class 0 mod d; the Farey page derives it from the automaton's stationary law under gcd(d, q Delta_F) = 1, Delta_F the gcd of the differences of the digits, states it as a conjecture and reads it to three decimals on eleven coprime moduli of base 3 {0,1}. This section proves that it is not a uniform upper bound.

Theorem 6.1 (the repunits). At base 3 with F = {0, 1} let R_t = (3^t - 1)/2, the number written with t ones, t >= 2. Then R_t lies in S_F, is coprime to 3, and

N_F(3^(2t); R_t) = 2^t + 1 ,    N_F(3^(3t); R_t) = 2 * 3^t + 1 ,    A_F(3^(2t)) = 4^t ,    A_F(3^(3t)) = 8^t .

Hence N_F(Q; R_t) R_t / A_F(Q) equals (2^t + 1)(3^t - 1)/(2 * 4^t), of order (3/2)^t, at Q = 3^(2t) with R_t < Q^(1/2), and (2 * 3^t + 1)(3^t - 1)/(2 * 8^t), of order (9/8)^t, at Q = 3^(3t) with R_t < Q^(1/3). No bound N_F(Q; d) <= C A_F(Q)/d holds uniformly over the d <= Q^(1/2) coprime to the base, nor over the d <= Q^(1/3).

Proof. R_t = 1 mod 3, so gcd(R_t, 3) = 1, and its digits are ones. 3^t = 1 mod R_t because 3^t - 1 = 2 R_t. A member n of S_F with 1 <= n <= 3^(kt) is either 3^(kt) itself, which is 1 mod R_t and not counted, or n = n_0 + n_1 3^t + ... + n_(k-1) 3^((k-1)t) with each n_i a padded string of t digits from {0, 1} read as an integer, and then n = n_0 + ... + n_(k-1) mod R_t. Write that sum as sum_(i < t) c_i 3^i with c_i the number of strings carrying a 1 at position i, 0 <= c_i <= k; it lies in [0, k R_t], so R_t | n exactly when it equals j R_t = sum_(i < t) j 3^i for some 0 <= j <= k, and j = 0 is n = 0, excluded. For k = 2: reducing mod 3 gives c_0 = j and then, dividing by 3 and inducting on t, every c_i = j. So j = 1 is the 2^t pairs with exactly one 1 in every position and j = 2 the single pair of two all-ones strings: N_F(3^(2t); R_t) = 2^t + 1. For k = 3 the same reduction gives c_i = j for j = 1, 2, which are 3^t triples each; for j = 3, c_0 is 0 or 3, and c_0 = 0 would put sum_(i >= 1) c_i 3^(i-1), a number of t - 1 digits at most 3 each and so at most (3^t - 3)/2, equal to R_t, impossible, so c_0 = 3 and induction gives every c_i = 3, the single triple of all-ones strings: N_F(3^(3t); R_t) = 2 * 3^t + 1. The counts A_F(3^(kt)) = 2^(kt) are the 2^(kt) - 1 nonzero strings of at most kt digits together with 3^(kt). The ratios follow, and R_t < 3^t = Q^(1/2) at k = 2, R_t < Q^(1/3) at k = 3. □

The Farey page carries the same count at Q = 3^(2rt) for every r and proves there, with a Perron root certified exactly at r <= 5, that the failure reaches every level Q^(1/(2r)) down to Q^(1/10); that argument is not reproduced here. What Theorem 6.1 costs is the step from (SR) to (U'): the uniform law N_F(Q; d) << A_F(Q)/d_co would make (SR) read |M_F(Q/d; d)| << d_co^(-1/2) Q^(alpha/2+eps), which is (U') up to the base-smooth factor.

Fact 6.2. The counts read N_F(3^(2t); R_t) = 5, 9, 17, 33, 65, 129, 257 for t = 2 to 8 and N_F(3^(3t); R_t) = 19, 55, 163, 487 for t = 2 to 5; over every d <= Q^(1/2) at Q = 3^8 to 3^16 the ratio N_F(Q; d) d_co/A_F(Q) is largest at the repunit of length ceil(L/2) at every level L, from 2.6562 at d = 40 to 12.8625 at d = 3280, and at those cells the Mobius values of the dilate stay under 0.870 of N_F^(1/2) in absolute partial sum. Domain: base 3 {0,1}, levels 8 to 16. Generator: lab/py/restricted-franel, verb accepting.

The readings

All numbers in this section are printed by uv run python research/lab/py/restricted-franel/restricted_franel.py sandwich and ... accepting, run from the repository root.

Fact 7.1 (the sandwich read). At base 3 {0,1}, Q = 3^4 to 3^8, and on the full set at Q = 40, 81, 243, the Jordan form of Lemma 4.1 equals the gcd double sum as an exact rational at every Q <= 729 and to 2.8e-14 and 5.7e-14 in floating point at Q = 2187 and 6561. With X = sum_(d <= Q) x_d^2 and Y = sum_(f <= Q) y_f^2:

Table 1. G_F/Y sits inside [6/pi^2, 1] = [0.607927, 1] at all eight values of Q, as Lemma 4.1 says it must; G_F/X sits inside the corridor [(6/pi^2)/N_Q, N_Q] of Theorem 4.3, [0.0436, 13.94] at Q = 81 and [0.0169, 35.95] at Q = 6561; no constant is claimed for either. The d = 1 share x_1^2/X is under a quarter throughout, so the dilates at d > 1 carry most of the mean square. The last two columns are the l^2 norms of the two maps of Theorem 4.3 on the Q x Q matrices, Rayleigh quotients after 3000 power steps and so lower bounds, against N_Q^(1/2) = 3.733351 to 5.995693 at Q = 81 to 6561: both grow and both sit under the bound. Generator: verb sandwich.

Fact 7.2 (the surrogate read). At base 3 {0,1}, Q = 3^4 to 3^12, B(Q) reads 200.233, 571.647, 1583.160, 4043.971, 10001.178, 23553.362, 54650.025, 125375.511, 285237.429, the Jordan form agreeing with the gcd double sum to 2.4e-11 at Q <= 3^8; B(Q)/sum tau(n) climbs from 2.9019 to 4.4174, the middle bound of Theorem 5.2 over B(Q) from 2.728 to 5.402, and B(Q)/(Q^alpha (ln Q)^2) falls from 0.6480 to 0.4007, no exponent claimed. Generator: verb accepting.

The figure is the same computation at Q = 81 run by its own binary, paper-franel-converse: the design's sixteen denominators from mrlynum::design::elements, the full set's nodes from the mediant walk mrlynum::lattice::farey and again from a gcd sweep, the Mobius values from mrlynum::factor::mobius_sieve. It asserts m_F(81) = 241, m(81) = 2020, sum_j delta_j^2 = 0.004002105 and 0.006524654 to 1e-9, M_F(81) = -2, M(81) = -4, sum_d x_d^2 = 19, G_F(81) = 12.574087 and G(81) = 159.157609 to 1e-5 against Table 1, and the rank form on both sets to 1e-8.

Open problems

Three things are open. First, (SR) itself: square-root cancellation of mu in every dilate against its own mass, which at d = 1 is the square-root conjecture for the design's Mertens sum and at every d off the base-power ladder a statement about a regular language that is not a digit design; Theorem 5.3 says it is enough and Theorem 4.5 says only its mean square is needed, and nothing here proves either; it is a conjecture. Second, the true order of the operator norms of the two maps: Proposition 4.4 puts the ratio of the mean squares above ln ln Q on some vectors and Theorem 4.3 with Robin's bound puts every ratio below O(ln Q ln ln Q), and the gap is untouched. Third, the accepting-set law at fixed coprime d, the Farey page's conjecture, which Section 6 refutes only as a uniform bound; on the strict set, both numerator and denominator in the design, there is no Franel-type statement to make until its limit measure is named, since at base 3 {0,1} the interval [1/2, 2/3] holds none of its fractions at any Q.

Reproducibility

One study, lab/py/restricted-franel, prints every number of Sections 6 and 7 and the constants of Section 3: uv run python research/lab/py/restricted-franel/restricted_franel.py sandwich from the repository root, one core, between 2.5 and 4.7 seconds on the two machines it was timed on, prints Table 1, the rank form's constant on six designs and the norms; ... accepting, about 12 seconds, prints Fact 7.2 and the counts of Fact 6.2; neither takes a further argument, reads a file or writes one, and the study's README names the witness of every printed line. Its other verbs check the frequency-m formula against the literal sum of roots of unity and the Fourier form against the truncated Fourier side with a printed tail bound; their readings are on the Farey page. The figure is bash scripts/figures.sh paper-franel-converse, under half a second a theme, with every drawn quantity asserted inside the binary as listed in Section 7.

References

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