This is too long for a comment.

There is something that I do not understand (more than surely, I am missing a point) : if I perform these linear regressions, I do not obtain the results given in the post.

Forcing $\beta=1$, what I obtain for the successive values of $j$ is $$\begin{array}{c|c}&j&1&2&3&4&5&6&7&\\\hline&\alpha&3.30&5.97&8.81&11.36&13.80&16.23&18.68&\end{array}$$ which looks like a quadratic in $\log(j)$.

Just to clarify, ig give below the syntax I used for genarating the data for Mathematica

  `data[j_]:=Table[{Prime[2k-1],Prime[2k]},{k,1,10^j}]`

This is too long for a comment.

There is something that I do not understand (more than surely, I am missing a point) : if I perform these linear regressions, I do not obtain the results given in the post.

Forcing $\beta=1$, what I obtain for the successive values of $j=10^i$ is $$\begin{array}{c|c}&j&1&2&3&4&5&6&7&\\\hline&\alpha&3.30&5.97&8.81&11.36&13.80&16.23&18.68&\end{array}$$ which looks like a quadratic in $\log(j)$.

Just to clarify, ig give below the syntax I used for genarating the data for Mathematica

  `data[j_]:=Table[{Prime[2k-1],Prime[2k]},{k,1,10^j}]`

Edit

Doing the same with $5^j$ instead of $10^j$ as before

$$\begin{array}{c|c}j&1&2&3&4&5&6&7&8&9&10&11\\\hline\alpha&2.60&3.88&6.34&8.41&9.90&11.79&13.55&15.24&16.94&18.66&20.36\end{array}$$