Empress Elisabeth of Austria did maths .

She is famous for this integral :

$$\int_{0}^{\pi}Si(x)Si(x)dx$$

Problem:
Let: $P=\frac{2}{3^{2}}+\frac{5}{2\cdot7^{2}}+\frac{11}{5\cdot13^{2}}+\frac{17}{11\cdot19^{2}}+\frac{23}{17\cdot29^{2}}+\frac{31}{23\cdot37^{2}}+\cdots+\frac{157}{149\cdot163^{2}}+\frac{167}{157\cdot173^{2}}$.

$P=\frac{2}{3^{2}}+\sum_{n=1}^{37}\frac{p_{n+2}}{p_np^2_{n+3}}$, where $p_n$ is the n th prime number.

Then show: $\left|\frac{\ln\left(P\right)}{\pi^{2}}\right|>0.12345678910\cdots$.