What is so interesting about the zeroes of the Riemann $\zeta$ function? The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$

The Riemann hypothesis asserts that all the non-trivial zeroes of the $\zeta$ function  lie on the line $\text{Re}(s) = \frac{1}{2}$. 

My question is:

Why are we interested in the zeroes of the $\zeta$ function? Does it give any information about something?

What is the use of writing $$\zeta(s) = \prod_{p} \biggl(1-\frac{1}{p^s}\biggr)^{-1}$$

RH has a direct influence on the chaotic quantum oscillator. Inverted chaotic quantum oscillator has eigen-energies equal to imaginary parts of the riemann zeta non-trivial roots, provided RH is correct. One can refer to http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry154.pdf for starters. Linear quantum chaotic oscillator being a fundamental object for a quantum theory in general, such as, say, linear harmonic quantum oscillator is, one is led to conclude therefore that RH plays a significant role in the breakthrough in this region.
Regards.
