Pinned primary sources

Every formula this study implements was read in at least one primary source during the session of 2026-08-17; the two load-bearing statements (Vasyunin's formula, the BCF asymptotic) were each pinned in two independent sources whose normalizations were checked against each other symbol by symbol before any code was written.

1. The criterion

• Nyman 1950, Beurling 1955 (via the surveys below): RH iff chi_(0,1) lies in the closed span of {t -> {theta/t}, 0 < theta <= 1} in L^2(0, infty) (equivalent forms on L^2(0,1) exist).
• Baez-Duarte, "A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis", Rend. Mat. Acc. Lincei 14 (2003) 5-11. arXiv: https://arxiv.org/abs/math/0202141 Integer dilations suffice: RH iff d_N -> 0 with d_N^2 = inf_a || chi_(0,1) - sum_{k=1}^N a_k rho(1/(kt)) ||^2_{L^2(0,infty)}, rho = fractional part. This is the d_N computed here.

2. Vasyunin's formula (the exact Gram entries)

• V. I. Vasyunin, "On a biorthogonal system associated with the Riemann hypothesis", Algebra i Analiz 7 (1995); St. Petersburg Math. J. 7 (1996) 405-419. Original statement (not directly consulted; quoted by the two sources below, whose statements agree).
• S. Bettin, J. B. Conrey, "Period functions and cotangent sums", Algebra and Number Theory 7 (2013) 215-242, arXiv:1111.0931. https://arxiv.org/abs/1111.0931 (eq. (13) and surrounding text, read in the ar5iv rendering https://ar5iv.labs.arxiv.org/html/1111.0931 and the arXiv PDF). With the Vasyunin sum V(h/k) = sum_{m=1}^{k-1} {m h / k} cot(pi m / k), gcd(h,k)=1, eq. (13) states, for coprime h, k >= 1: (1/(2 pi sqrt(hk))) Int_{-infty}^{infty} |zeta(1/2+it)|^2 (h/k)^{it} dt/(1/4+t^2) = (log 2pi - gamma)/2 * (1/h + 1/k)
• ((k-h)/(2hk)) log(h/k)
• (pi/(2hk)) ( V(h/k) + V(k/h) ). Also from the same paper: c_0(h/k) = -sum_{m=1}^{k-1} (m/k) cot(pi m h/k) and V(h/k) = -c_0(hbar/k), h hbar == 1 (mod k); the reciprocity formula for c_0 is theirs and is not used here.
• B. Landreau, F. Richard, "Le critere de Beurling et Nyman pour l'hypothese de Riemann: aspects numeriques", Experiment. Math. 11 (2002) 349-360. PDF: https://projecteuclid.org/journals/experimental-mathematics/volume-11/issue-3/Le-crit%C3%A8re-de-Beurling-et-Nyman-pour-lhypoth%C3%A8se-de-Riemann/em/1057777427.pdf Their Theoreme 2.1 (attributed to Vasyunin) gives, for e_n(t) = {1/(nt)}, omega = gcd(n,m), n = omega n0, m = omega m0: <e_n, e_m> = (log 2pi - gamma)/2 * (1/n + 1/m)
• ((m-n)/(2mn)) log(n/m)
• (pi omega/(2nm)) sum_{j=1}^{n0-1} {j m0/n0} cot(pi j/n0)
• (pi omega/(2mn)) sum_{j=1}^{m0-1} {j n0/m0} cot(pi j/m0). Cross-check performed: dividing the Bettin-Conrey coprime statement by omega and simplifying reproduces the Landreau-Richard statement exactly (the mixed-argument form used in the code). Mellin route from |zeta|^2 integral to the inner product: <e_j, e_k> equals the eq. (13) integral by Mellin-Plancherel with Mellin(e_k)(s) = -zeta(s) k^{-s}/s on Re s = 1/2.

3. The target inner products (derived, then validated)

b_k = <chi_(0,1), e_k> = (1 - gamma + log k)/k. Derivation: substitute u = 1/(kt), use int_1^infty {u}/u^2 du = 1 - gamma and int_{1/k}^1 du/u = log k. Landreau-Richard state the equivalent <chi, g_theta> = theta(1 - gamma - log theta) for theta = 1/k (their section 2). Validated here by direct quadrature (see RESULTS.md). Special case: <e_1, e_1> = log 2pi - gamma, so d_1^2 = 1 - (1-gamma)^2/(log 2pi - gamma) = 0.85821205139551088467... (closed form, digits computed at 40 dps in this session).

4. The conjectured asymptotic and its status

• Baez-Duarte, Balazard, Landreau, Saias, "Notes sur la fonction zeta de Riemann, 3", Adv. Math. 149 (2000) 130-144: unconditional lower bound liminf d_N^2 log N >= sum_{Re rho = 1/2} 1/|rho|^2 (distinct zeros), and the conjecture d_N^2 ~ C/log N.
• J.-F. Burnol (cited in BCF as [Bur]): improved lower bound liminf d_N^2 log N >= sum_{Re rho = 1/2} m(rho)^2/|rho|^2.
• S. Bettin, J. B. Conrey, D. W. Farmer, "An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion", arXiv:1211.5191, Proc. Steklov Inst. Math. 2013. https://arxiv.org/abs/1211.5191 Theorem 1: if RH holds and sum_{|Im rho| <= T} 1/|zeta'(rho)|^2 << T^{3/2-delta} for some delta > 0, then the smoothed Mobius choice V_N(s) = sum_{n<=N} (1 - log n/log N) mu(n) n^{-s} achieves (1/2pi) Int |1 - zeta V_N(1/2+it)|^2 dt/(1/4+t^2) = (1/log N) sum_rho 1/|rho|^2 + O(1/log^2 N), hence d_N^2 ~ (2 + gamma - log 4pi)/log N if all zeros are simple (sum over distinct zeros; under RH, sum_rho 1/|rho|^2 with rho and 1-rho paired equals 2 + gamma - log 4pi = 0.04619141793...). The displayed error term O(1/log^2 N) is read directly from the end of their proof of Theorem 1 ("moving the line of integration in (7) ... the integral is equal to (1/log N) sum_rho 1/|rho|^2 + O(1/log^2 N)"). So the predicted next-order shape tested in this study is d_N^2 log N = C + O(1/log N), C = 2 + gamma - log 4pi. Note their theorem is an upper-bound statement about the specific vector V_N; the true minimizer satisfies d_N^2 <= that value, and the Burnol bound pins the constant from below.

5. Prior numerical values (validation targets)

• Landreau-Richard 2002 (same PDF as above), the only substantial published float computation found: d_n by Gram-Schmidt with exact Vasyunin entries up to n = 20000 (confirmed by conjugate gradient and QR up to n = 10000). No table of d_n digits is printed; the paper gives (a) Figure 1: d_n decreasing from ~0.10 to ~0.066 over n = 1..20000, with the window [0.07, 0.08] crossed during n ~ 2000..10000 (Figure 2); (b) the least-squares fit d_n ~ a_N/sqrt(log n) with a_N ~= 0.21377 at N = 20000, against sqrt(2+gamma-log 4pi) = 0.21492 (their section 3); (c) Conjecture 3.1 restated: d_n^2 ~ (2+gamma-log 4pi)/log n. These are the external anchors reproduced in RESULTS.md.
• Maslanka's mid-2000s numerics (searched 2026-08-17): what was located concerns the Riesz/Baez-Duarte c_k moment criterion, not d_N; no d_N table found there, so no Maslanka validation target is used.
• Repo-internal: data/baez_duarte_gram_N50_dps50.json and zeta.criteria.baez_duarte_d2 compute the BBLS basis A_k = {1/(kx)} - (1/k){1/x} on L^2(0,1), k = 2..N, target 1. That is a different quadratic form with the same conjectured d_N^2 log N limit; it is compared, not equated, in RESULTS.md.

6. Constants

gamma = 0.5772156649015328606..., log 2pi = 1.8378770664093454836..., C = 2 + gamma - log 4pi = 0.0461914179322420676286204958134... (matches zeta.criteria.BD_CONSTANT, itself pinned by the repo's tests as 2*lambda_1 of Li's criterion).