Why are these integrals equivalent to the Riemann Hypothesis?

Question

Riemann Hypothesis is equivalent to the integral equation

$$\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt =0$$

Many other integral equations exist that are equivalent.

How to show that they are equivalent ?

They usually include absolute value of a function.

Why is that ?

I assume it came from a contour integral on the riemann sphere.

The growth rate of the zeta function on the critical line also probably relates to all those integrals , not ?

Another example is

Establishing the exact value $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$ is equivalent to the Riemann Hypothesis.

More examples :

Riemann's Hypothesis is true if and only if $$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(\frac{1}{2}+it)}{\zeta(\frac{1}{2})}\right|\ \frac{dt}{t^2}=\frac{\pi}{8}+\frac{\gamma}{4}+\frac{\log 8\pi}{4}-2$$

Take $a\in R$ with $\frac{1}{2}\leq a<1$. Riemann's $\zeta$-function has no zeros in $\Re(s)>a$ if and only if $$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(a+it)}{\zeta(a)}\right|\ \frac{dt}{t^2}=\frac{\zeta'(a)}{2\zeta(a)}-\frac{1}{1-a}$$

And many more exist.

I have no idea how to get to such conclusions or prove them.

Im not even sure how to prove integrals for "slightly easier" cases , meaning not famous open problems but zero's of other nontrivial functions that are not on a half-plane.