The geometry of zeta values: what each value counts, and how its continued fraction moves

SUMMARY A zeta value is usually met as a number in a formula: π²/6 in the Sommerfeld expansion, ζ(3) in the number of photons per volume, ζ(3/2) in the critical density of an ideal Bose gas, 1/120 in the Casimir force between plates. This paper asks a different question: what does each value count, and how does it unfold when it is computed step by step? The aim is an arithmetic explanation of known physics, not a new prediction. For each of the values ζ(1/2), ζ(3/2), ζ(2), ζ(5/2), ζ(3), ζ(4), ζ(5), ζ(0), ζ(−1/2), ζ(−1), ζ(−3), ζ′(0), ζ′(−1) and for the pole at s = 1 the paper gives, as far as they exist, four readings:• a building block: n^(−s) as the reciprocal of a geometric object at step n (a hypotenuse √n of the spiral of Theodorus for s = 1/2, a cube of edge √n for s = 3/2, a shell of a lattice);• a lattice: a density (visible points of ℤ^k have density 1/ζ(k)), a lattice sum (Madelung constant, Epstein zeta function) or the first shell of a lattice;• a wave: the signed error of an expansion, either the regular continued fraction, the Euler continued fraction built from the building blocks, or Apéry's continued fractions with polynomial partial quotients;• a meaning: the physical law in which the value appears.Almost every identity used is classical and is labelled as such. The contribution is the arrangement, the figures, a companion script that checks every number, and four observations that we have not found stated together. MAIN OBSERVATIONS 1. Euler's continued fraction of the shells (Proposition 5.2, proved). For s > 0 and a_n = n^s, η(s) = 1/(a₁ + a₁²/((a₂ − a₁) + a₂²/((a₃ − a₂) + a₃²/((a₄ − a₃) + …)))), the n-th convergent equals the partial sum Σk≤n (−1)^(k+1) k^(−s), and the error R_n alternates in sign with ½(n+1)^(−s) ≤ |R_n| ≤ ½n^(−s). The partial denominators are the shells n^s − (n−1)^s: all equal to 1 for s = 1 (the classical log 2 = 1/(1 + 1²/(1 + 2²/(1 + 3²/…)))), the odd numbers (gnomons of the squares) for s = 2, the centred hexagonal numbers 3n² − 3n + 1 for s = 3, and the shells of a chain of cubes of edge √n for s = 3/2, where ζ(3/2) = η(3/2)/(1 − 2^(−1/2)). The error is a regular wave of height ½ in units of the n-th building block (Figure 5). 2. Two kinds of continued fraction (Sections 6 and 7, proved, cited and numerical). The regular continued fractions of ζ(1/2), …, ζ(5) behave like those of a typical real number: over 300 partial quotients (1/n) log q_n lies between 1.15 and 1.23 against Lévy's constant π²/(12 log 2) = 1.1866, and the share of partial quotients 1 lies between 0.39 and 0.47 against the Gauss–Kuzmin value 0.4150. Apéry's continued fractions are structured: the error for ζ(2) shrinks per step by φ^(−10) = 0.0081306, for ζ(3) by (1+√2)^(−8) = 0.00086655, i.e. by powers of the fundamental units of ℚ(√5) and ℚ(√2). The code measures 0.0081324 at depth 45 and 0.00086719 at depth 30. 3. First shells from ζ(0) (Proposition 8.1 and Remark 8.2, proved). For ℤ⁴, D₄ and E₈ the lattice zeta function is a product of two Riemann zeta functions, Zℤ⁴(s) = 8(1 − 4^(1−s))ζ(s)ζ(s−1), ZD₄(s) = 24·2^(−s)(1 − 2^(1−s))ζ(s)ζ(s−1), ZE₈(s) = 240·2^(−s)ζ(s)ζ(s−3). If Z_Λ(s) = N c(s) ζ(s) ζ(s−k+1) with a finite Dirichlet series c(s) normalised as in the paper, the classical value Z_Λ(0) = −1 gives N = −1/(c(0)ζ(0)ζ(1−k)): 240 = −1/(ζ(0)ζ(−3)) = −8/B₄ for E₈, 24 for D₄, 8 for ℤ⁴. By von Staudt–Clausen the denominator of B₄ = −1/30 is 2·3·5 = 30, which is also the Coxeter number of E₈; the exponents of E₈ are 1, 7, 11, 13, 17, 19, 23, 29, the eight units modulo 30. The same 30 indexes the cyclotomic ring ℤ[ζ₃₀] of rank φ(30) = 8 over which E₈ is a free module of rank one, and the residue class of a prime modulo 30 decides the law of the first shell of its prime ideals (Krause, First Shells of Prime Ideals in Cyclotomic Lattices, https://doi.org/10.5281/zenodo.23293397). The coincidence h(E₈) = 30 = denominator(B₄) is recorded, not explained (open). 4. A neutral lattice and a single wall (Section 8, proved, cited; reading as hypothesis). For every lattice Λ ⊂ ℝ^d, Z_Λ(0) = −1 and Z_Λ(−m) = 0 for m = 1, 2, …: with the origin added, the regularised number of all points and every regularised moment vanish. On the full line ℤ this is the Poisson summation formula, the remainder is O(e^(−π²N²)). On the half-line ℕ the sum Σ n^k f(n/N) leaves the constant ζ(−k) for every smooth cut-off f with f(0) = 1: −1/2, −1/12, 0, 1/120, 0, −1/252, … We read the non-positive values as what a single wall leaves of a neutral lattice: a Bernoulli number for odd k, nothing for even k ≥ 2. In this sense ζ(−k) acts as a neutraliser that compensates the divergent volume term; Casimir-type forces appear only where the lattice is cut. This is a reading of regularisation and proposes no new force. FURTHER CONTENT • The wheel modulo 30 (Section 3): removing the Euler factors at 2, 3, 5 leaves the sum over the eight spokes, W(s) = ζ(s)(1−2^(−s))(1−3^(−s))(1−5^(−s)); W(0) = 0, W(−1) = 2/3; visible points of ℤ^k have density 1/ζ(k).• The spiral of Theodorus and ζ(1/2) (Section 5.1): Σk≤N 1/√k = 2√N + ζ(1/2) + 1/(2√N) + O(N^(−3/2)); the total angle of the spiral is 2√N + K + …, with K = Σ (−1)^m ζ(m+1/2)/(2m+1) = −2.1577829966… (Brink 2012); the Madelung constant of the square lattice is 4(√2 − 1)ζ(1/2)β(1/2) = −1.6155426267…; ζ(−1/2) = −ζ(3/2)/(4π) is the reflection partner.• The cube chain and ζ(3/2), ζ(5/2) (Section 5.2): ζ(3/2) = Σ 1/(√n)³ is the sum of the fractions of the n-th cube filled by the unit cube, and at the same time the critical phase-space density of the ideal Bose gas; Bloch's T^(3/2) and T^(5/2) laws for magnons.• Integer values (Section 6): ζ(2) in the Sommerfeld expansion and the density of visible points, ζ(3) in photon number and the Bose gas in a harmonic trap, ζ(4) in the Stefan–Boltzmann and Debye laws, 5!ζ(5) = 124.4313… in the Bloch–Grüneisen T⁵ law.• Derivatives (Section 8.4): ζ′(0) = −½ log 2π (regularised product of all positive integers √(2π), functional determinants), ζ′(−1) = 1/12 − log A.• An overview table of the four readings for every value (Section 9) and a table of the classical sources (Section 11). FIGURES (black and white, English labels) 1. ζ(s) on the real line with its physical anchors. 2. Wheel modulo 30, visible points of ℤ², share of the wheel. 3. First shells in a Coxeter plane: 16-cell (8), 24-cell (24), E₈ (240 roots, eight rings of 30). 4. Spiral of Theodorus and the convergence to ζ(1/2) and K. 5. Cube chain and the error wave of Euler's continued fraction. 6. Apéry's continued fractions against regular continued fractions (logarithm of the error). 7. Error waves q_n(q_n x − p_n) of seven zeta values and of φ². 8. Full line against half-line; remainders ζ(−k). NUMERICAL VERIFICATION The companion script zv_checks.py performs 31 checks (sections A–G), zv_figures.py redraws the eight figures and checks the 9 numbers shown in them (H1–H9). Last full run: 31 ok and 9 ok, 0 failed, about 25 s on two cores. Selected results: Σk≤N 1/√k − 2√N − 1/(2√N) = −1.4603545089 at N = 10⁶ against ζ(1/2) = −1.4603545088; the Madelung theta integral and the closed form agree to 15 digits; shell numbers of ℤ⁴ (Jacobi, n ≤ 3000), D₄ (m ≤ 30) and E₈ (norms 2, 4, 6: 240, 2160, 6720) agree exactly with the divisor-sum formulas; Z_Λ(0) = −1 for ℤ², ℤ⁴, D₄, E₈; Euler's continued fraction reproduces the partial sums of η(s) for s = 1/2, 1, 3/2, 2, 3; the number of roots equals rank × Coxeter number for B₄, D₄, F₄, E₈ with h computed from the highest root. STATEMENTS THAT FAIL (negative) • ζ(2) = [1; 1, 1, 1, 4, 2, …] and ζ(3/2) = [2; 1, 1, 1, 1, 2, …] begin like φ = [1; 1, 1, …] and φ² = [2; 1, 1, …], but ζ(2) − φ = 0.0269 and ζ(3/2) − φ² = −0.0057; there is no golden structure in the regular expansion beyond the first steps. STATUS OF THE RESULTS Every result in the paper carries one label: proved; proved, cited (classical statements with reference); classical physics; numerical; hypothesis (the author's geometric readings); open; negative. No identity is used to predict a physical constant, and nothing is claimed about irrationality, transcendence or the zeros of ζ. Open questions: a continued fraction for ζ(5) or ζ(3/2) whose error contracts by a fixed unit; an explanation of h(E₈) = denominator(B₄); a geometric object of the cube-chain type whose constant term is ζ(3/2) alone, as ζ(1/2) is the constant term of the root spiral; a physical system with a cut-off on the wheel modulo 30 in which W(−1) = 2/3 appears. The identification of the partial denominators of Euler's continued fraction with the shells of a √n-cube chain, the reading of Proposition 8.1 as a normalisation of the first shell by Z_Λ(0) = −1, and the side-by-side comparison of Apéry's contraction rates with the units of ℚ(√5) and ℚ(√2) were not found in the literature searched (arXiv, the journals cited and general web search); all three are elementary consequences of the cited results. A systematic search of the review databases is still to be done. - zeta values - - - - - -