gaussian-franel

• Franel's identity one field up: an exact formula for the Fourier L^2 discrepancy of the Gaussian Farey set on the torus C/Z[i], written as a gcd-weighted double sum over Gaussian Mertens sums.
• The node set is the one spun-stack lights: w = u/d in lowest terms in Z[i], denominators one per associate class with N(d) <= N, numerators the reduced residues mod d, read modulo Z[i] in the unit square.
• Three results: the Gaussian Ramanujan sum in Mobius form (Theorem 1), the identity itself (Theorem 2), and the two-sided equivalence with the Riemann hypothesis for zeta_{Q(i)}(s) = zeta(s) L(s, chi_-4) (Theorem 3).
• Everything exact where the claim is exact: Gaussian Euclidean gcd, complete residue systems on the gcd box, sums of roots of unity reduced modulo the cyclotomic polynomial, and the identity itself an exact rational.

THE OBJECT

• Pairing: <lambda, w> = Re(lambda w), which is p x - q y for lambda = p + qi and w = x + iy. It is Z[i]-periodic in w, so e_lambda(w) = e(<lambda, w>) is a character of C/Z[i], and lambda -> e_lambda is an isomorphism of Z[i] onto the dual group.
• The choice matters once. Under this pairing the character e_lambda is trivial on (1/d) Z[i] / Z[i] exactly when d | lambda; under the Euclidean pairing Re(conj(lambda) w) the same condition reads conj(d) | lambda. The two differ by relabelling the dual by conjugation, which fixes N(lambda), so no printed number below depends on the choice.
• The node set: G_N = {u/d mod Z[i] : [d] an associate class with N(d) <= N, u mod d, gcd(u, d) = 1}, of size m = sum_{[d], N(d) <= N} Phi(d), the Gaussian totient sum. This is the lit set of the exact spun stack.
• Against the complex Farey set of arXiv:2407.04380, G_T = {pr(p/q) : p, q in Z[i], 0 < |q| <= T}: the two sets are equal, with N = T^2, the factor being 1. Letting q run over all four associates and p over all of Z[i] produces each torus point exactly once anyway, since a reduced denominator class [d] contributes the Phi(d) points u/d however its associates are listed. Verified by building G_T literally, q over every element of norm at most T^2 and p over the box [0, N(q))^2, which covers a complete residue system because N(q) and i N(q) both lie in the ideal: the two sets agree at T = 2, 3, 4, 5, 6 with 4, 24, 64, 176, 320 points (check_sayous).

THE FUNCTIONAL

• There is no rank. Franel's sum_j delta_j^2 needs the linear order of F_Q on [0,1] and C/Z[i] has none, so the twin is a choice and is named as one.
• The choice: D_2(N)^2 = (1/m^2) sum_{lambda != 0} |S_N(lambda)|^2 / N(lambda)^2 with S_N(lambda) = sum_{w in G_N} e(<lambda, w>), where N(lambda) = |lambda|^2.
• Why this one. Writing nu_N for the deviation of the empirical measure from Haar, the sum is || nu_N * K ||_2^2 for the kernel with hat K(lambda) = N(lambda)^-1, so D_2(N) = 4 pi^2 || U_N ||_2 with U_N the periodic Newtonian potential of nu_N, -Laplacian U_N = nu_N. It is translation-invariant, invariant under the unit group of Z[i], needs no fundamental domain and no anchored box, and it metrises weak-* convergence to Haar, so it is a Weyl criterion in L^2. The anchored L^2 star-discrepancy fails all four.
• The classical twin of the same functional is Edwards' I. With S(k) = sum_{rho in F_Q} e(k rho) and B1bar the sawtooth, int_0^1 (sum_rho B1bar(u + rho))^2 du = (1/(4 pi^2)) sum_{k != 0} |S(k)|^2 / k^2, so the classical m^2 D_2^2 is 4 pi^2 I, and Edwards section 12.2 evaluates I as sum_{a,b} M(Q/a) M(Q/b) gcd(a,b)^2 / (12 a b). Theorem 2 is that evaluation one field up.

THEOREM 1, THE EXPONENTIAL SUM

• Partition G_N by reduced denominator class: S_N(lambda) = sum_{[d], N(d) <= N} c_d(lambda) with the Gaussian Ramanujan sum c_d(lambda) = sum_{u mod d, gcd(u,d) = 1} e(<lambda, u/d>). Well defined: u -> u + dt moves u/d by t in Z[i] and <lambda, t> is an integer; replacing d by an associate permutes the reduced residues, so c_d depends only on the ideal.
• Kluyver in Z[i]. c_d(lambda) = sum_{e | gcd(d, lambda)} mu_G(d/e) N(e), the sum over ideal divisors, mu_G the Mobius function on ideals of Z[i], with the convention gcd(d, 0) = d so that c_d(0) = Phi(d). Proved.
• Proof. Every residue u mod d has gcd(u, d) = d/e for a unique ideal e | d, and then u/d = u'/e with u' reduced mod e, so sum_{e | d} c_e(lambda) = sum_{u mod d} e(<lambda, u/d>). The right side is the sum of a character over the finite abelian group Z[i]/(d); that character is trivial iff lambda/d in Z[i], tested on u = 1 and u = i, so the sum is N(d) when d | lambda and 0 otherwise. Mobius inversion over the divisor lattice of the ideal (d), which is a lattice of ideals because Z[i] is a principal ideal domain, gives the stated formula.
• The Mertens form. Writing d = e f and M_G(x) = sum_{[f], N(f) <= x} mu_G(f) for the Gaussian Mertens function over associate classes, S_N(lambda) = sum_{[e] | lambda, N(e) <= N} N(e) M_G(N / N(e)). Proved, by exchanging the two sums. At lambda = 0 this is the zero mode m = sum_{[e], N(e) <= N} N(e) M_G(N/N(e)), and at any unit lambda it is S_N(lambda) = M_G(N).
• Verified exactly at N = 50. All 2720 Gaussian Ramanujan sums, one for each of the 40 associate classes and each of the 68 nonzero lambda with N(lambda) <= 20, are computed by literal summation over the reduced residues as integer vectors of roots of unity reduced modulo Phi_{N(d)}, and every one is a rational integer equal to the Mobius formula: 0 mismatches. The 68 class sums equal the Mertens form: 0 mismatches. The 68 literal sums over all 672 nodes agree with both in floating point, worst error 1.281e-13 (check_theorem_1).
• Verified, the zero mode: 120, 672, 10608 at norm bounds 20, 50, 200, read both as sum Phi(d) and as sum N(e) M_G(N/N(e)) (main). The middle two are the node counts spun-stack prints.

THEOREM 2, FRANEL ONE FIELD UP

• The identity. With F(N) = sum_{[a],[b] : N(a), N(b) <= N} (N(gcd(a,b))^2 / (N(a) N(b))) M_G(N/N(a)) M_G(N/N(b)),

m^2 D_2(N)^2 = 4 zeta_K(2) F(N),    zeta_K(2) = zeta(2) L(2, chi_-4) = zeta(2) * Catalan = 1.506703

• Proved. Substitute the Mertens form of Theorem 1, expand the square, and exchange: for fixed ideals a, b the inner sum is sum_{lambda != 0, lcm(a,b) | lambda} N(lambda)^-2 = 4 zeta_K(2) / N(lcm(a,b))^2, the 4 because every nonzero ideal has four generators. Then N(a) N(b) / N(lcm(a,b))^2 = N(gcd(a,b))^2 / (N(a) N(b)). The double sum is finite because M_G(N/N(a)) vanishes for N(a) > N, so the right side is exact, and F(N) is an exact rational.
• The kernel is the gcd matrix N(gcd(a,b))^2 / (N(a) N(b)), the Gaussian twin of the gcd(m,n)^2/(mn) that is the layer Gram matrix of the parity carpet stack on the stack page. The same Smith kernel carries the moire correlation law and the Franel identity.
• Verified at N = 20 and N = 50 against the Fourier side truncated at N(lambda) <= 200000, the tail bounded by m^2 (4 zeta_K(2) - sum_{0 < N(lambda) <= 200000} N(lambda)^-2) since |S_N(lambda)| <= m. At N = 20, m = 120, F(N) = 114917096/3663075 = 31.371756, identity 189.071678 against truncated Fourier side 189.069795, gap 0.001883 inside the tail bound 0.226192. At N = 50, m = 672, F(N) = 17870882021826065419/177236423132266875 = 100.830753, identity 607.687997 against 607.677451, gap 0.010546 inside the tail bound 7.093392 (check_theorem_2).
• The classical control is the same code path on the rational data. At Q = 40 the direct Farey enumeration gives m = 490 and sum delta_v^2 = 0.0104270117, and C(Q) - 1 = 12 m sum delta_v^2 holds as an identity of exact rationals, which is Edwards section 12.2 regenerated here; the sieve route reads C(40) = 62.310828579 against the exact 62.310828579 (classical_exact, farey_delta_square, franel_form). Verified.

THEOREM 3, THE EQUIVALENCE

• Statement. F(N) = O(N^{1+eps}) for every eps > 0, equivalently m^2 D_2(N)^2 = O(N^{1+eps}), equivalently D_2(N) = O(N^{-3/2+eps}), is equivalent to the Riemann hypothesis for zeta_{Q(i)}(s) = zeta(s) L(s, chi_-4). That hypothesis is RH and GRH for chi_-4 together and is strictly stronger than RH; the product is named every time it appears and is never shortened.
• The load-bearing input is M_G(x) = O(x^{1/2+eps}) for every eps > 0 if and only if zeta_K has no zero with Re s > 1/2. It is not called standard here; both directions are stated with their hypotheses.
• The input, backward. The bound gives the hypothesis. Proved, elementarily. Write m(n) = sum_{N(a) = n} mu_G(a), so M_G(x) = sum_{n <= x} m(n) and sum_n m(n) n^{-s} = 1/zeta_K(s) on Re s > 1. Partial summation gives sum_{n <= x} m(n) n^{-s} = M_G(x) x^{-s} + s int_1^x M_G(t) t^{-s-1} dt, and under the bound both pieces converge as x -> infinity whenever Re s > 1/2. A Dirichlet series is analytic in its half plane of convergence, so 1/zeta_K continues analytically to Re s > 1/2, and a zero of zeta_K there would be a pole of it. Nothing about zeta_K beyond its Euler product is used.
• The input, forward. The hypothesis gives the bound. Proved, by Littlewood's argument transcribed to zeta_K, with the two analytic inputs cited rather than assumed. Perron at half-integer x writes M_G(x) = (1/(2 pi i)) int_{Re s = 1 + 1/log x} x^s / (s zeta_K(s)) ds up to the standard truncation error, and the contour is pushed left to Re s = 1/2 + eps and closed on horizontal segments at heights where zeta_K is not small. Both inputs are proved for a Dedekind zeta in arXiv:2109.06665, On a Mertens-type conjecture for number fields, Math. Proc. Cambridge Philos. Soc., read at source. Its Lemma 5.4 gives, under the hypothesis and for |t| >= 1, |log zeta_K(s)| <= n_K (log(1/(1 - sigma)) + O((log tau)^{2 - 2 sigma} / ((1 - sigma) log log tau))) on 1/2 + 1/log log tau <= sigma <= 1 - 1/log log tau with tau = |t| + 4, which exponentiates to 1/zeta_K(sigma + it) << tau^delta for every delta > 0, uniformly on sigma >= 1/2 + eps. Its Lemma 2.4 gives, under the same hypothesis, heights T_n in [n, n+1) with |zeta_K(sigma + i T_n)| >= exp(-C log n / log log n) for -1 <= sigma <= 2, which are the horizontal segments the contour closes on. Taking T = x^2 makes each of the three pieces O(x^{1/2 + eps + delta}), so the bound holds at half-integers and hence everywhere, M_G moving between consecutive half-integers by the number of ideals of one norm, which is O(x^eps).
• The biconditional is assembled here, not quoted. That paper states neither half as an equivalence; what it supplies is its (1.1), 1/zeta_K(s) = s int_1^infinity M_K(x) x^{-s-1} dx, which is the backward direction's Mellin identity one field up, its Lemma 5.4 and its Lemma 2.4, and the two paragraphs above are the assembly. The argument's shape is Edwards section 12.1, read at source, which supplies the completion Littlewood omitted; that section cites Titchmarsh Theorem 14.25(c) territory for the rational biconditional and is K = Q only.
• Theorem 3, forward. The hypothesis gives the decay. Proved, through the forward input above. Under |M_G(x)| <= C_eps x^{1/2+eps}, each term of Theorem 1 is at most C_eps N^{1/2+eps} N(e)^{1/2}, and only e with N(e) <= min(N, N(lambda)) contribute, so |S_N(lambda)| <= C_eps N^{1/2+eps} d(lambda) min(N, N(lambda))^{1/2} with d the number of ideal divisors. Splitting the lambda sum at N(lambda) = N gives sum_{N(lambda) <= N} d(lambda)^2 / N(lambda) = O(log^4 N) and N sum_{N(lambda) > N} d(lambda)^2 / N(lambda)^2 = O(log^3 N), so m^2 D_2(N)^2 = O(N^{1+3eps}).
• Theorem 3, backward. The decay gives the hypothesis. Proved outright, the backward input being elementary. In one line: the four units lambda have N(lambda) = 1 and S_N(lambda) = M_G(N), so dropping every other term of the Fourier sum gives M_G(N)^2 <= zeta_K(2) F(N). Hence F(N) = O(N^{1+eps}) forces M_G(N) = O(N^{1/2+eps}), and the backward input turns that into the absence of zeros right of the critical line.
• The two halves are not equally cheap. Backward needs nothing but the Euler product and partial summation; forward needs Perron and two conditional estimates for zeta_K, both read at source in the paper cited above. The backward direction is easier than Franel's because the functional is a Fourier sum by construction while sum_j delta_j^2 is a rank statistic. What is not free is the identity: the content is Theorem 2, which turns a two-dimensional discrepancy into a finite gcd-weighted quadratic form in Gaussian Mertens sums, and the threshold N^{1+eps}, which is derived rather than inherited.
• The inequality of the backward direction is Verified at every rung of the meter, M_G(N)^2 / (zeta_K(2) F(N)) never rising above 0.017 (main).
• What the equivalence is not. The Gaussian Farey set on C/Z[i] is not the rational Farey sequence carrying a character weight. Huxley's theorem for lambda(q) = chi(q) reaches the zeros of one Dirichlet L-function through weighted rational Farey points; this object is unweighted, two-dimensional, and invariant under the unit group, and it reaches zeta and L(s, chi_-4) together because zeta_K factors. Whether a number-field Farey sequence in the announced second part of that paper already carries a discrepancy statement is not settled here, that part being unread.

WHAT IT PRINTS

• check_sayous at T = 2, 3, 4, 5, 6: the literal complex Farey set G_T and the node set at N = T^2 are equal, with 4, 24, 64, 176, 320 points.
• check_residues to norm bound 200: 158 associate classes, 0 residue systems of the wrong size, 0 collisions mod d, 0 disagreements with the Gaussian totient. The residues of d are the box [0, g) x [0, N(d)/g) with g = gcd(Re d, Im d), a fundamental domain of the ideal because its area is the index and its two sides are the Hermite normal form of the lattice.
• check_theorem_1 at N = 50 and N(lambda) <= 20: 672 nodes, 68 characters, 2720 exact Gaussian Ramanujan sums, 0 Mobius-formula mismatches, 0 Mertens-form mismatches, 0 literal node-sum mismatches at 1e-9, worst literal error 1.281e-13.
• main, the zero mode: sum Phi(d) and sum N(e) M_G(N/N(e)) both read 120, 672, 10608 at norm bounds 20, 50, 200.
• readout: sum_{N(a) <= x} M_G(x/N(a)) = 1 at every x from 1 to 2000, the Gaussian twin of the collapsing global readout of the Mertens meter.
• check_theorem_2 at N = 20, 50: the exact rational F(N), the identity, the Fourier side truncated at N(lambda) <= 200000, the gap and the printed tail bound, with the gap inside the bound at both.
• farey_delta_square and classical_exact at Q = 40: m = 490, sum delta_v^2 = 0.0104270117, and C(Q) - 1 = 12 m sum delta_v^2 as exact rationals.
• franel_form on the rational data at Q = 125 to 8000: S2 * Q = (C(Q) - 1) Q / (12 Phi(Q)) reads 0.5395, 0.5848, 0.6241, 0.6387, 0.6560, 0.6538, 0.6564, regenerating the Farey discrepancy table of the Farey page with no Farey enumeration anywhere.
• franel_form on the Gaussian data at N = 100 to 64000: the meter table, m, F(N), F(N)/N, D_2(N)^2 N^3, the local log-log slope of F, M_G(N), the ratio M_G(N)^2 / (zeta_K(2) F(N)), and the classical C(N)/N beside it.

• Reading the table: the local slope of F walks to 1.0042 and F(N)/N climbs slowly from 2.12 to 3.02, which is the shape a bounded power of a logarithm has and is what the equivalence predicts; the classical column climbs the same way. No exponent is claimed beyond this window, which is 10 nested points and cannot separate N^{1+eps} from N^{1.02}. Verified, as a window.

RUN

• uv run python research/lab/py/gaussian-franel/gaussian_franel.py
• From the repository root. One core, under one second.
• Domain: the identification with the complex Farey set at T = 2 to 6, residue systems to norm bound 200, exact checks at norm bound 50 with N(lambda) <= 20, the identity at norm bounds 20 and 50 against the Fourier side to N(lambda) <= 200000, the classical control at Q = 40 and the classical meter to Q = 8000, the Gaussian meter to N = 64000.
• Nothing is written to disk.

WITNESSES

• Theorem 1: ramanujan_exact against ramanujan_formula and sum_formula, with literal_sum over the node set as the third route.
• Theorem 2: franel_exact against lambda_side, with the printed tail bound; farey_delta_square against classical_exact as the published control.
• Theorem 3: the meter columns of main, and M_G(N)^2 / (zeta_K(2) F(N)) <= 1 at every rung.
• The node set and its count: node_set and totient_class, agreeing with the Gaussian totient sums of spun-stack; check_residues for the residue systems and check_sayous for the identification with the complex Farey set of the literature.