Bound for the number of roots $\rho$ of $\xi(\rho)$

I was reading the book Riemann's Zeta Function, by H. M. Edwards, page 42, where is a theorem that estimates the number of roots of the $$\xi$$ function $$\xi(s)=\Gamma\Big(\frac{s}{2}+1\Big)(s-1)\pi^{-s/2}\zeta(s),$$ built with the Gamma function and Riemann's Zeta Function. The estimation is inside or on the circle $$|s-\frac{1}{2}|=R$$. Below is the screenshot of the theorem and the proof.

There is one step of this proof that I don't understand. How is proven the last inequality? The one that establish that $$\frac{2}{\log 2}R\log R+ 2R-\frac{\log\xi(\frac{1}{2})}{\log 2}\leq 2R\log R.$$

• At first this is a mistake. Do you really care of the constant at this point ? The theorem about the density of zeros is proven in every text about $\zeta(s)$. – reuns Jan 9 at 21:32
• @reuns I wouldn't care about the constant, if it can be shown that the order of the number of roots is at most $R\log R$. Can you give me a reference to find an alternative proof to this fact? – Julián Jan 9 at 22:19
• It should be in your book. The density theorem is $\sim$ not $\le$ and it follows those lines, otherwise look in Titchmarsh – reuns Jan 9 at 22:55
• I got it. The proof given by Edwards shows under the table that the order of the number of roots is at most $R\log R$. Thank you for your clarification @reuns – Julián Jan 14 at 2:07