This dataset accompanies the preprint: T. Toguchi, "Numerical observations on extremal Fourier coefficients of the Syracuse random variable, " 2026. Abstract: For the Syracuse random variable Syrac (Z/3ⁿ Z) introduced by Tao (Forum of Mathematics, Pi, 2022), we report numerical computations of its characteristic function for 1 <= n <= 21, covering supports of size up to 3²1 ≈ 1. 05 × 10¹0. Writing psiₙ (k): = Eexp (-2*pi*i * 2ᵏ * Syrac / 3ⁿ) and k* (n): = argmaxₖ |psiₙ (k) | in a canonical fundamental domain, we observe that k* (n) exhibits an approximately linear drift with slope near 4/3. The simple closed-form predictor floor (4 (n-1) /3) matches k* (n) for n in 4, 5,. . . , 17 and for n = 19, but fails at n in 18, 20, 21, with k* (n) - floor (4 (n-1) /3) equal to +1, +1, and +2 respectively. The ratio k* (21) /21 equals 4/3 exactly. The interval slope (k* (21) - k* (4) ) /17 = 24/17 ≈ 1. 412 slightly exceeds 4/3 because of the endpoint deviation at n = 21. We pose the asymptotic behaviour of k* (n) as a numerical open question. Repository: https: //github. com/toguchitaku/syrac-extremal-fourier Contents: - preprint. pdf, preprint. tex: paper- preprintdataₙ21. json: numerical data underlying Table 1- finaldata. py, finaldataₕighres. py: reference implementations- verifyₚsi. py: independent verification script License: MIT for code and data; CC BY 4. 0 for the preprint (see README. md in the repository).
Taku Toguchi (Mon,) studied this question.