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| Formaat: | Recurso digital |
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Zenodo
2026
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| Online toegang: | https://doi.org/10.5281/zenodo.19699245 |
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- <p>For the Wigner surmise family p_β(s) = a_β s^β exp(-c_β s^2) at arbitrary<br>Dyson index β > 0, the Master Formula of [Paper IV, 10.5281/zenodo.19667720]<br>gives κ_n(log s_β) = (-1)^n (n-1)! / 2^n · ζ_H(n, (β+1)/2) for n ≥ 2.<br>We prove that for every rational β = p/q with gcd(p,q) = 1, the Hurwitz<br>zeta value on the right-hand side admits a unique Q-linear decomposition<br>into Dirichlet L-values L(n, χ) attached to real characters χ modulo m,<br>where m is the reduced denominator of (β+1)/2. A parity duality governs<br>tractability: L(n, χ) reduces to a rational multiple of π^n/sqrt(|d_χ|)<br>(via generalised Bernoulli numbers) iff the parity of χ matches that of<br>n; otherwise L(n, χ) is generically transcendental (e.g., Catalan's<br>constant G = L(2, χ_{-4}) for χ odd, n even).</p> <p>As corollaries we obtain closed forms for κ_n(β) at<br>β ∈ {1/2, 3/2, 1/3, 2/3, 1/4, 3/4} for small n, including<br>κ_2(β = 3/2) = π²/4 + 2G - 4,<br>κ_3(β = 2/3) = -91 ζ(3)/4 + π³√3/2,<br>κ_7(β = 1/2) = 61π⁷/4 - 45720 ζ(7),<br>to the author's knowledge not previously assembled in the literature.<br>All identities verified to at least 30 decimal digits.</p> <p>Companion note to [Paper IV, 10.5281/zenodo.19667720], addressing §7(v).</p>