On the 'reverse theorem' for functional equation of the Riemann zeta function

Question

The following is the first part of Section 2.13. of Titchmarsh's book The Theory of the Riemann Zeta-Function:

My first question (orange-underlined) is: How $R_{\nu} = x^{-\frac12 s_{\nu}} Q_{\nu}(\log x)$ holds? I mean, why we can write each $Q_{\nu}(x)$ as a polynomial with indeterminate $\log x$?

The rest of the section is as follows:

My second question (blue-underlined) is: how to prove that $\int_0^{\infty} x^a \log^b(x) e^{ - \pi t^2 x} dx$ is a sum of the terms of the form $t^{\alpha} \log^{\beta}(x)$? Again, WolframAlpha couldn't help!

Question Three (pink-underlined): The LHS includes $-\pi \sum_{k=1}^{m} t^{\alpha_k +1} \log^{\beta}(x).$ How replacing $t$ with $t+i$ leaves the expression unchanged?

Answer 1

1. The $\log x$ would come into play if you'd have a pole of order $2$ or greater. So basically to get to $R_1$ you expand everything into power series around $s_1$ and multiply out and collect the $-1$ coefficient, so you could get $\log x$ terms from $x^{-s/2}=e^{-\frac {\log x}{2}s}$.

2. Just substitute to pull out the $t's$ from the integral.

3. The LHS should be periodic, not each term of the LHS. Think about what you would get if you'd replace $f$ with $\zeta$.