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. enter image description here

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

  • $\begingroup$ 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 $\zeta(s)$. $\endgroup$ – reuns Jan 9 at 21:32
  • $\begingroup$ @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? $\endgroup$ – Julián Jan 9 at 22:19
  • $\begingroup$ It should be in your book. The density theorem is $\sim$ not $\le $ and it follows those lines, otherwise look in Titchmarsh $\endgroup$ – reuns Jan 9 at 22:55
  • $\begingroup$ 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 $\endgroup$ – Julián Jan 14 at 2:07

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