Bound for the number of roots $rho$ of $xi(rho)$
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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
$$xi(s)=GammaBig(frac{s}{2}+1Big)(s-1)pi^{-s/2}zeta(s),$$
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
$$frac{2}{log 2}Rlog R+ 2R-frac{logxi(frac{1}{2})}{log 2}leq 2Rlog R.$$
complex-analysis number-theory riemann-zeta riemann-hypothesis
asked Jan 9 at 20:51
JuliánJulián
826
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add a comment |
$begingroup$
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
$$xi(s)=GammaBig(frac{s}{2}+1Big)(s-1)pi^{-s/2}zeta(s),$$
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
$$frac{2}{log 2}Rlog R+ 2R-frac{logxi(frac{1}{2})}{log 2}leq 2Rlog R.$$
complex-analysis number-theory riemann-zeta riemann-hypothesis
asked Jan 9 at 20:51
JuliánJulián
826
$endgroup$
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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
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@reuns I wouldn't care about the constant, if it can be shown that the order of the number of roots is at most $Rlog R$. Can you give me a reference to find an alternative proof to this fact?
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– Julián
Jan 9 at 22:19
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It should be in your book. The density theorem is $sim$ not $le $ and it follows those lines, otherwise look in Titchmarsh
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– reuns
Jan 9 at 22:55
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I got it. The proof given by Edwards shows under the table that the order of the number of roots is at most $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07
add a comment |
$begingroup$
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
$$xi(s)=GammaBig(frac{s}{2}+1Big)(s-1)pi^{-s/2}zeta(s),$$
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
$$frac{2}{log 2}Rlog R+ 2R-frac{logxi(frac{1}{2})}{log 2}leq 2Rlog R.$$
complex-analysis number-theory riemann-zeta riemann-hypothesis
asked Jan 9 at 20:51
JuliánJulián
826
$endgroup$
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
$$xi(s)=GammaBig(frac{s}{2}+1Big)(s-1)pi^{-s/2}zeta(s),$$
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
$$frac{2}{log 2}Rlog R+ 2R-frac{logxi(frac{1}{2})}{log 2}leq 2Rlog R.$$
complex-analysis number-theory riemann-zeta riemann-hypothesis
complex-analysis number-theory riemann-zeta riemann-hypothesis
asked Jan 9 at 20:51
JuliánJulián
826
asked Jan 9 at 20:51
JuliánJulián
826
asked Jan 9 at 20:51
JuliánJulián
826
asked Jan 9 at 20:51
JuliánJulián
826
asked Jan 9 at 20:51
JuliánJulián
826
826
$begingroup$
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 $Rlog R$. 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 $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07
add a comment |
$begingroup$
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 $Rlog R$. 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 $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07
$begingroup$
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
$begingroup$
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 $Rlog R$. Can you give me a reference to find an alternative proof to this fact?
$endgroup$
– Julián
Jan 9 at 22:19
$begingroup$
@reuns I wouldn't care about the constant, if it can be shown that the order of the number of roots is at most $Rlog R$. 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
$begingroup$
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 $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07
$begingroup$
I got it. The proof given by Edwards shows under the table that the order of the number of roots is at most $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07
add a comment |
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$begingroup$
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 $Rlog R$. 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 $Rlog R$. Thank you for your clarification @reuns
$endgroup$
– Julián
Jan 14 at 2:07