Shintani’s prehomogeneous zeta functions and multiple sine functions
Copyright policy )Shintani’s prehomogeneous zeta functions and multiple sine functions
For \(n\geq 1\) let \(Z_n(s)\) be Shintani's prehomogeneous zeta function associated to the space of symmetric matrices. The author proves that for \(n\equiv 1\pmod 4\), \(n\neq 1\), the function \( Z_n(s)\) has a simple zero at \(s=0\), and \[ Z_n'(0)=-(-4)^{\frac{1-n}{4}}(\zeta(-1)\zeta(-3)\dots \zeta(-(n-2)))^2\log\left(\prod_{k=1}^{\frac{n-1}{4}}S_{\frac{n+1}{2}}(k)^{a(\frac{n+1}{2},k)}\right), \] where \(S_r(x)\) is the multiple sine function and \(a(2m+1,k)\) are certain numbers explicitly defined in the paper.
Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Shintani's prehomogeneous zeta function, \(\zeta (s)\) and \(L(s, \chi)\), multiple sine function
Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Shintani's prehomogeneous zeta function, \(\zeta (s)\) and \(L(s, \chi)\), multiple sine function
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