Engineering Mathematics: Numerical value of Riemann zeta function (Euler Product)

Saturday, October 20, 2012

NVRZ.ID.1: Numerical value of Euler Product

$\prod \left ( \frac{1+p^{-2s}}{1-p^{-2s}} \right )= \frac{5}{2}\pi^{2}$

refer to RS.ID.2

$\frac{\zeta ^{2}(s)}{\zeta (2s)}= \prod \left ( \frac{1+p^{-s}}{1-p^{-s}} \right )$

(1)

if s=2 , then

$\frac{\zeta ^{2}(2)}{\zeta (4)} = \prod \left ( \frac{1+p^{-2s}}{1-p^{-2s}} \right )$

(2)

$\zeta (2)= \frac{\pi ^{2}}{6}$

$\zeta (4)= \frac{\pi ^{2}}{90}$

(3)

From (2) and (3) ,

$\frac{\frac{\pi ^{2}}{6}\frac{\pi ^{2}}{6}}{\frac{\pi ^{2}}{90}} = \prod \left ( \frac{1+p^{-2s}}{1-p^{-2s}} \right )$

(4)

$\frac{5}{2}\pi^{2} = \prod \left ( \frac{1+p^{-2s}}{1-p^{-2s}} \right )$

(5)

Engineering Mathematics