Divergent Products over non-trivial zeros of the Riemann zeta function

SPECIES

$\displaystyle \prod_{\rho} 1 = 1 \quad \text{(convergent)}$

$\displaystyle \prod_{\rho} \rho = \xi(0) = \dfrac{1}{2}$

$\displaystyle \prod_{\rho} (\rho - 1) = \xi(1) = \dfrac{1}{2}$

$\displaystyle \prod_{\rho} \left(1 - \dfrac{1}{\rho}\right) = \dfrac{\xi(1)}{\xi(0)} = 1$

$\displaystyle \prod_{\rho} (\rho + 1) = \xi(-1) = \xi(2) = \dfrac{\Gamma(2)\,\zeta(2)}{\pi} = \dfrac{\pi}{6}$

$\displaystyle \prod_{\rho} (\rho - 2) = \xi(2) = \dfrac{\Gamma(2)\,\zeta(2)}{\pi} = \dfrac{\zeta(2)}{\pi} = \dfrac{\pi}{6}$

$\displaystyle \prod_{\rho} (\rho + 2) = \xi(-2) = \xi(3) = \dfrac{2\,\Gamma(\frac{5}{2})\,\zeta(3)}{\pi^\frac{3}{2}} = \dfrac{3\,\zeta(3)}{2\pi}$

$\displaystyle \prod_{\rho} (\rho - 3) = \xi(3) = \dfrac{2\,\Gamma(\frac{5}{2})\,\zeta(3)}{\pi^\frac{3}{2}} = \dfrac{3\,\zeta(3)}{2\pi}$

$\displaystyle \prod_{\rho} (\rho + 3) = \xi(-3) = \xi(4) = \dfrac{3\,\Gamma(3)\,\zeta(4)}{\pi^2} = \dfrac{\zeta(4)}{\zeta(2)} = \dfrac{\pi^2}{15}$

$\displaystyle \prod_{\rho} (\rho - 4) = \xi(4) = \dfrac{3\,\Gamma(3)\,\zeta(4)}{\pi^2} = \dfrac{\zeta(4)}{\zeta(2)} = \dfrac{\pi^2}{15}$

$\displaystyle \prod_{\rho} (\rho + 4) = \xi(-4) = \xi(5) = \dfrac{4\,\Gamma(\frac{7}{2})\,\zeta(5)}{\pi^\frac{5}{2}} = \dfrac{15\,\zeta(5)}{2\pi^2}$

$\displaystyle \prod_{\rho} (\rho - 5) = \xi(5) = \dfrac{4\,\Gamma(\frac{7}{2})\,\zeta(5)}{\pi^\frac{5}{2}} = \dfrac{15\,\zeta(5)}{2\pi^2}$

$\displaystyle \prod_{\rho} (\rho + 5) = \xi(-5) = \xi(6) = \dfrac{5\,\Gamma(4)\,\zeta(6)}{\pi^3} = \dfrac{2\pi^3}{63}$

Pages: 1 2