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 $$\zeta(s) = \prod_{p} \biggl(1-\frac{1}{p^s}\biggr)^{-1}$$

the HIlbert Polya operator which would prove Riemann Hypothesis is the Wu-Sprung genralzed model with potential
$$  f^{-1} (x)=\frac{4}{\sqrt{4x+1} } +\frac{1}{4\pi } \int\nolimits_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right)  -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right)  $$
with boundary conditions $$ y(0)=0=y(\infty) $$ and $ H= -\frac{d^{2}}{dx^{2}}y(x)+f(x)y(x)=E_{n}$...................$E_{n}=\gamma_{n}^{2}$·
however mathematician do not like it, if we take the half derivative then we find the distributional Riemann-Weil formula for the zeros
$$ \begin{array}{l}
\sum\limits_{n=0}^{\infty }\delta \left( x-\gamma _{n} \right) + \sum\limits_{n=0}^{\infty }\delta \left( x+\gamma _{n} \right)  =\frac{1}{2\pi } \frac{\zeta }{\zeta } \left( \frac{1}{2} +ix\right) +\frac{1}{2\pi } \frac{\zeta '}{\zeta } \left( \frac{1}{2} -ix\right) -\frac{\ln \pi }{2\pi }  \\[10pt]
{} +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} -i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{1}{\pi } \delta \left( x-\frac{i}{2} \right) +\frac{1}{\pi } \delta \left( x+\frac{i}{2} \right) \end{array} $$
