Rational-β Wigner surmise cumulants: a Dirichlet L-function decomposition
Fuente:
Zenodo
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Recurso digital |
| Publié: |
Zenodo
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866901680862265344 |
|---|---|
| author | Niedbala Giraudin, David |
| author_facet | Niedbala Giraudin, David |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19699245 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Rational-β Wigner surmise cumulants: a Dirichlet L-function decomposition Niedbala Giraudin, David Wigner surmise, Hurwitz zeta, Dirichlet L-functions, random matrix theory, rational β, generalised Bernoulli numbers, Catalan constant, β-ensembles <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> |
| title | Rational-β Wigner surmise cumulants: a Dirichlet L-function decomposition |
| topic | Wigner surmise, Hurwitz zeta, Dirichlet L-functions, random matrix theory, rational β, generalised Bernoulli numbers, Catalan constant, β-ensembles |
| url | https://doi.org/10.5281/zenodo.19699245 |