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Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$ where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$ Note that $I$ converges ...

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...