totient-constant

• The node count of the spun stacks is a totient sum over ideals, and this study proves its constant instead of fitting it.
• It serves lab/py/spun-stack (the Gaussian stack, node count 672, 10608, 168088 at norm bounds 50, 200, 800) and lab/py/eisenstein-stack (the hexagonal stack, 630, 9606, 151020, 337026, 945486, 2419950 at 50, 200, 800, 1200, 2000, 3200), whose limit constants were Conjecture and are now Proved.
• Everything is recounted from scratch here, twice: once by enumerating associate classes and factoring norms, once by the norm-indexed Dirichlet series, which is the route that also carries class number above one.

THE THEOREM

• Let K be an imaginary quadratic field with ring of integers O_K, class number h, unit group of order w, and discriminant D_K. Write N for the ideal norm and Phi(a) = |(O_K/a)^*| = N(a) prod_{p | a} (1 - 1/N(p)).
• Proved: sum_{N(a) <= N} Phi(a) = (rho_K/(2 zeta_K(2))) N^2 + O(N^(3/2)) with rho_K = 2 pi h/(w sqrt |D_K|) the ideal density, the sum running over nonzero ideals.
• At h = 1 nonzero ideals are nonzero associate classes, which is the spun stacks' index set.
• Proved: for K = Q(i), h = 1, w = 4, |D_K| = 4, the constant is pi/(8 zeta(2) G) = 0.260634696495 with G Catalan's constant.
• Proved: for K = Q(sqrt -3), h = 1, w = 6, |D_K| = 3, the constant is pi/(6 sqrt 3 zeta(2) L(2, chi_-3)) = 0.235217881630.
• The error O(N^(3/2)) is the one derived below and is weaker than the two-signed deviations the two studies print, which are of size N log N; those deviations sit inside the theorem and are not claimed to be its true order.

CONVENTIONS

• The sum is over nonzero integral ideals throughout; at class number one the unit group acts freely on nonzero elements, each class holds exactly w elements, and ideals and associate classes are the same monoid.
• Phi is the ideal totient, the order of the unit group of O_K/a, and it is multiplicative because the Chinese remainder theorem splits O_K/a.
• mu_K is the ideal Mobius function, (-1)^k on a squarefree product of k prime ideals and 0 otherwise.
• The Gaussian window is a + b i with a >= 1 and b >= 0, one representative per class; the Eisenstein window is a + b omega minimal in its orbit under the six units, omega acting as (a, b) -> (-b, a - b) and -1 as (a, b) -> (-a, -b).
• Norm bound N throughout, never a radius: the height |q| <= T of a complex Farey fraction is the norm bound N = T^2.

THE IDEAL COUNT

• Lemma, Proved: A(t) = #{a : N(a) <= t} = rho_K t + E(t) with |E(t)| = O(sqrt t) and rho_K = 2 pi h/(w sqrt |D_K|).
• First the principal ideals. O_K is a rank two lattice in C of covolume sqrt |D_K|/2, so counting its points in the disc of radius R = sqrt t by attaching to each point its fundamental cell, of diameter delta, gives cells contained in the disc of radius R + delta and covering the disc of radius R - delta; dividing the two areas by the covolume gives 2 pi t/sqrt |D_K| + O(sqrt t) lattice points, and removing the origin and dividing by w gives 2 pi t/(w sqrt |D_K|) + O(sqrt t) principal ideals, with an effective implied constant, 4 pi delta/(w sqrt |D_K|) at leading order.
• Then one class at a time. Fix an ideal class C and an integral ideal a in C^-1; the map b -> a b = (x) is a bijection from the integral ideals of C of norm at most t to the nonzero elements of a of norm at most t N(a), taken up to units, since (x) a^-1 is integral exactly when x lies in a.
• a is a lattice of covolume N(a) sqrt |D_K|/2, so the same cell argument gives 2 pi t/(w sqrt |D_K|) + O(sqrt t) ideals in every class alike, and summing the h classes gives the lemma.
• This is the Gauss circle count for Z[i] and the hexagonal count for Z[omega], whose closed forms sum_j (floor(t/(4j+1)) - floor(t/(4j+3))) and sum_j (floor(t/(3j+1)) - floor(t/(3j+2))) are Verified in lab/py/spun-stack and lab/py/eisenstein-stack; the lemma above is derived here from the lattice and cites neither.
• rho_K is also the residue of zeta_K at s = 1 given by the class number formula, which is the cross-check: pi/4 for Q(i) and pi/(3 sqrt 3) for Q(sqrt -3).

THE CONVOLUTION STEP

• The method is Dirichlet convolution followed by Abel summation, not a hyperbola split: the whole range of the ideal b is summed and the inner sum is estimated uniformly.
• Proved: Phi = N * mu_K as a Dirichlet convolution on ideals, since both sides are multiplicative and on a prime power (N * mu_K)(p^k) = N(p)^k - N(p)^(k-1) = Phi(p^k); equivalently sum_a Phi(a) N(a)^-s = zeta_K(s - 1)/zeta_K(s).
• Hence S(N) = sum_{N(a) <= N} Phi(a) = sum_{N(b) <= N} mu_K(b) T(N/N(b)) with T(X) = sum_{N(c) <= X} N(c).
• Abel summation on the lemma gives T(X) = X A(X) - int_0^X A(u) du = (rho_K/2) X^2 + X E(X) - int_0^X E(u) du, so |T(X) - (rho_K/2) X^2| <= (5/3) C X^(3/2) with C the constant of the lemma.
• The main term is (rho_K/2) N^2 sum_{N(b) <= N} mu_K(b) N(b)^-2, and the tail sum_{N(b) > N} N(b)^-2 = O(1/N) by Abel summation on A(u) = O(u), so the main term is (rho_K/(2 zeta_K(2))) N^2 + O(N).
• The error term is at most (5/3) C N^(3/2) sum_{N(b) <= N} N(b)^(-3/2) <= (5/3) C zeta_K(3/2) N^(3/2), finite because zeta_K(s) converges for s > 1.
• Adding the two gives the theorem, with an effective implied constant (5/3) C zeta_K(3/2) + O(1).
• Proved, conditional refinement: if the lemma holds with E(t) = O(t^theta) for some 0 < theta < 1, the same three steps give error O(N^(1 + theta)); the elementary theta = 1/2 is the one derived here, and no sharper circle exponent is claimed. The lower constraint is not decoration: at theta = 0 the error layer needs zeta_K(1), which diverges, and the conclusion drops to O(N log N).

THE CONSTANTS

• zeta_K(s) = zeta(s) L(s, chi_D) for a quadratic field, chi_D the Kronecker symbol of the discriminant, so zeta_K(2) = zeta(2) G for Q(i) and zeta(2) L(2, chi_-3) for Q(sqrt -3).
• c = rho_K/(2 zeta_K(2)) = pi h/(w sqrt |D_K| zeta(2) L(2, chi_D)), which is pi/(8 zeta(2) G) and pi/(6 sqrt 3 zeta(2) L(2, chi_-3)).
• main prints, at 50 working digits through the Hurwitz form L(2, chi_D) = |D_K|^-2 sum_a chi_D(a) zeta(2, a/|D_K|) with Euler-Maclaurin and pi by Machin: zeta(2) = 1.644934066848, G = 0.915965594177, L(2, chi_-3) = 0.781302412896, zeta_K(2) = 1.506703009923 and 1.285190955484, rho_K = 0.785398163397 and 0.604599788078, and the constants 0.260634696495 and 0.235217881630.
• Those two agree with the values lab/py/spun-stack and lab/py/eisenstein-stack print, 0.260635 and 0.235217881630015, which are now derived rather than fitted.
• class_number derives h from the finite character sum h = w (-sum_a chi_D(a) a)/(2 |D_K|), returning 1, 1, 2 and 3 at D_K = -4, -3, -20, -23, so the constant is assembled from w, |D_K|, h and L(2, chi_D) alone.

THE GENERAL FIELD

• Proved: the theorem holds for every imaginary quadratic field, of any class number. The convolution step never sees the class number, and the lemma is proved above one ideal class at a time.
• Verified: at D_K = -20, h = 2, the constant is 0.378582556790 and the count at norm bound 102400 is 3969730264, ratio 1.000001116781; at D_K = -23, h = 3, the constant is 0.425700255391 and the count is 4463281160, ratio 0.999885848149 (norm_totient_sum).
• Proved as an implication only: for a general number field, with ideals throughout, the convolution step is unchanged and gives (rho_K/(2 zeta_K(2))) N^2 + O(N^(1 + theta)) for any ideal count A(t) = rho_K t + O(t^theta) with 0 < theta < 1; that ideal count is not derived here, so nothing about degree above two is claimed outright.

THE FAREY CONVENTION

• The lit set of the spun stack is the complex Farey set: Sayous arXiv:2407.04380 Section 2 defines F_t = {pr_{O_K}(p/q) : p, q in O_K, p O_K + q O_K = O_K, 0 < |q| <= e^(t/2)} inside C/O_K and states card F_t ~ c_K e^(2t) with c_K = pi/(sqrt |D_K| zeta_K(2)), attributed there to Cosentino, Ergodic Theory Dynam. Systems 19 (1999) 1437-1484, Theorem 4.
• Since |q| <= e^(t/2) is the norm bound N = e^t, the two conventions are: counting denominators as ideals gives pi/(w sqrt |D_K| zeta_K(2)) = rho_K/(2 zeta_K(2)), and counting them as elements gives pi/(sqrt |D_K| zeta_K(2)), exactly w times larger, w = 4 for Z[i] and 6 for Z[omega].
• Verified: farey_set builds the set F_t itself, layer by layer as the union of the lattices q^-1 O_K modulo O_K over all classes of norm at most N, deduplicating points in exact integer coordinates, and returns 672, 10608, 168088 for Z[i] and 630, 9606, 151020 for Z[omega] at norm bounds 50, 200, 800, equal in all six cases to the totient sum.
• F_t is written as a set of points of the torus, so its cardinality is the ideal one; the factor w is inherited from the cited Cosentino statement rather than chosen, and c_K e^(2t) is the size of the family indexed by element denominators, in which each point repeats once per associate of its denominator.

WHAT IT PRINTS

• classes enumerates one representative per nonzero associate class in each field, 80414 Gaussian and 61914 Eisenstein classes of norm at most 102400.
• totient computes Phi from the factorisation of the norm alone, splitting rational primes into ramified, inert and split, a split p contributing (1 - 1/p)^2 exactly when p divides both coordinates.
• totient_sum reads the prefix array built by tables and returns 672, 10608, 168088, 372872, 1045088, 2663864, 42663808, 683283504, 2732257376 for Q(i) and 630, 9606, 151020, 337026, 945486, 2419950, 38560362, 616981926, 2466558234 for Q(sqrt -3) at norm bounds 50, 200, 800, 1200, 2000, 3200, 12800, 51200, 102400, matching the published counts of lab/py/spun-stack and lab/py/eisenstein-stack at all nine bounds where they overlap.
• main prints count/(c N^2) at those bounds: 1.031329, 1.017516, 1.007684, 0.993494, 1.002445, 0.998113, 0.999097, 1.000066, 0.999746 for Q(i) and 1.071347, 1.020968, 1.003192, 0.995017, 1.004904, 1.004699, 1.000578, 1.000604, 1.000049 for Q(sqrt -3), oscillating about 1 rather than approaching it from one side.
• main prints the deviation scaled by the derived error, (count - c N^2)/N^(3/2), never larger than 0.119 in absolute value across both fields and all nine bounds, and the same deviation scaled by N log N, which reproduces the +0.214, +0.186, +0.090, -0.198, +0.304, +0.438 of lab/py/eisenstein-stack.
• convolution_sum re-evaluates the same totient sum as sum_{N(b) N(c) <= N} mu_K(b) N(c) from an independently computed mu_K, agreeing at all nine bounds in both fields, which is the numerical form of Phi = N * mu_K.
• farey_set counts the complex Farey set directly, 672, 10608, 168088 and 630, 9606, 151020 at norm bounds 50, 200, 800.
• norm_arrays and norm_totient_sum recount by norm alone, with no lattice: the ideals of norm n number sum_{d | n} chi_D(d), the ideal Mobius function by norm is mu * (mu chi_D) from 1/zeta_K = (1/zeta)(1/L), and S(N) = sum_k m(k) T(N/k). This agrees with the class route at all eighteen bounds of the two class-number-one fields and supplies the tables at D_K = -20 and -23.

RUN

• uv run python research/lab/py/totient-constant/totient_constant.py
• From the repo root. One core, about a second, standard library only.
• Domain is norm bounds up to 102400 for the counts, up to 800 for the direct Farey set, four discriminants for the general statement, and the constants at 50 working digits.
• Writes nothing.

WITNESSES

• The theorem's constant: main against the counts of totient_sum at nine norm bounds in both fields, with the ratio and both scaled deviations.
• The recount: totient_sum against the published counts of lab/py/spun-stack and lab/py/eisenstein-stack, nine matches, no mismatch.
• The convolution Phi = N * mu_K: convolution_sum against totient_sum, eighteen agreements.
• Class number above one: norm_totient_sum at D_K = -20 and -23, ratios 1.000001116781 and 0.999885848149 at norm bound 102400, with class_number deriving h = 2 and 3.
• The Farey convention: farey_set against totient_sum, six agreements, settling that the set convention carries rho_K/(2 zeta_K(2)) and the element convention w times that.