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Effectiveness of Landau's Prime Ideal Theorem

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Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same time, it seems hard to believe that an effective version in the style of Page's theorem on the prime numbers within $[1,x]$ in arithmetic progressions of length $ll (log x)^2-epsilon$ isn't possible.

I am asking for an effective proof of the asymptotic statement, not, as I've just said, of the best known error term.

analytic-number-theory

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asked Mar 29 at 3:44

NellNell

1033

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  You need effective bounds $|zeta_K(1+it)| ge A_K log^-1(2+|t|)$, $|zeta_K'(s)| le B_K log(2+|t|)^c_K$ for $Re(s) > 1-fracd_Klog(2+$ from which you obtain $|zeta_K(s)|ge E_K log(2+|t|)^-f_K $ for $Re(s) > 1-fracg_Klog^h_K(2+$. There you can apply the usual method for the PNT. You can look into Iwaniec-Kowalski they should give an effective PNT for most L-functions.
  $endgroup$
  – reuns
  Mar 29 at 7:00
  
  

  
  
  
  


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  I see that the crucial statement is Theorem 5.35 in Iwaniec-Kowalski, which states that the only exceptional zeroes there can be for Grossencharakters are Siegel zeroes of quadratic characters. The usual bounds on such zeroes can be plugged into Theoreö 5.33. It would still be satisfying to have a reference to a self-contained statement that would be completely clear to the non-specialist.
  $endgroup$
  – Nell
  Mar 29 at 13:47
  

  
  
  
  

add a comment | 

0

$begingroup$

Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same time, it seems hard to believe that an effective version in the style of Page's theorem on the prime numbers within $[1,x]$ in arithmetic progressions of length $ll (log x)^2-epsilon$ isn't possible.

I am asking for an effective proof of the asymptotic statement, not, as I've just said, of the best known error term.

analytic-number-theory

share|cite|improve this question

asked Mar 29 at 3:44

NellNell

1033

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You need effective bounds $|zeta_K(1+it)| ge A_K log^-1(2+|t|)$, $|zeta_K'(s)| le B_K log(2+|t|)^c_K$ for $Re(s) > 1-fracd_Klog(2+$ from which you obtain $|zeta_K(s)|ge E_K log(2+|t|)^-f_K $ for $Re(s) > 1-fracg_Klog^h_K(2+$. There you can apply the usual method for the PNT. You can look into Iwaniec-Kowalski they should give an effective PNT for most L-functions.
  $endgroup$
  – reuns
  Mar 29 at 7:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see that the crucial statement is Theorem 5.35 in Iwaniec-Kowalski, which states that the only exceptional zeroes there can be for Grossencharakters are Siegel zeroes of quadratic characters. The usual bounds on such zeroes can be plugged into Theoreö 5.33. It would still be satisfying to have a reference to a self-contained statement that would be completely clear to the non-specialist.
  $endgroup$
  – Nell
  Mar 29 at 13:47
  

  
  
  
  

add a comment | 

$begingroup$

Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same time, it seems hard to believe that an effective version in the style of Page's theorem on the prime numbers within $[1,x]$ in arithmetic progressions of length $ll (log x)^2-epsilon$ isn't possible.

I am asking for an effective proof of the asymptotic statement, not, as I've just said, of the best known error term.

analytic-number-theory

share|cite|improve this question

asked Mar 29 at 3:44

NellNell

1033

$endgroup$

share|cite|improve this question

asked Mar 29 at 3:44

NellNell

1033

share|cite|improve this question

asked Mar 29 at 3:44

NellNell

1033

asked Mar 29 at 3:44

NellNell

1033

asked Mar 29 at 3:44

NellNell

1033