Number of primes $p$ less than $n$ s.t $p equiv 3 pmod 4$












0












$begingroup$


I would like to find a lower and upper bound for the number of primes less than $n$ that are $3pmod 4$. My guess is that it should be close to half of $pi (n)$ but cannot find a proof or generate myself. Is there a way to prove some lower and upper bounds as we have for $pi (n)$?



Thanks






number-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
    $endgroup$
    – JavaMan
    Dec 18 '18 at 16:54








  • 1




    $begingroup$
    Related post. Also, see the Polya conjecture.
    $endgroup$
    – Arthur
    Dec 18 '18 at 16:55






  • 1




    $begingroup$
    Dirichlet's Theorem addresses this question.
    $endgroup$
    – lulu
    Dec 18 '18 at 16:55












  • $begingroup$
    Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
    $endgroup$
    – oren1
    Dec 18 '18 at 17:06








  • 1




    $begingroup$
    Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
    $endgroup$
    – Bruno Andrades
    Dec 18 '18 at 17:29


















0












$begingroup$


I would like to find a lower and upper bound for the number of primes less than $n$ that are $3pmod 4$. My guess is that it should be close to half of $pi (n)$ but cannot find a proof or generate myself. Is there a way to prove some lower and upper bounds as we have for $pi (n)$?



Thanks






number-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
    $endgroup$
    – JavaMan
    Dec 18 '18 at 16:54








  • 1




    $begingroup$
    Related post. Also, see the Polya conjecture.
    $endgroup$
    – Arthur
    Dec 18 '18 at 16:55






  • 1




    $begingroup$
    Dirichlet's Theorem addresses this question.
    $endgroup$
    – lulu
    Dec 18 '18 at 16:55












  • $begingroup$
    Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
    $endgroup$
    – oren1
    Dec 18 '18 at 17:06








  • 1




    $begingroup$
    Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
    $endgroup$
    – Bruno Andrades
    Dec 18 '18 at 17:29
















0












0








0





$begingroup$


I would like to find a lower and upper bound for the number of primes less than $n$ that are $3pmod 4$. My guess is that it should be close to half of $pi (n)$ but cannot find a proof or generate myself. Is there a way to prove some lower and upper bounds as we have for $pi (n)$?



Thanks






number-theory







share|cite|improve this question











$endgroup$




I would like to find a lower and upper bound for the number of primes less than $n$ that are $3pmod 4$. My guess is that it should be close to half of $pi (n)$ but cannot find a proof or generate myself. Is there a way to prove some lower and upper bounds as we have for $pi (n)$?



Thanks






number-theory




number-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 18 '18 at 20:28









rtybase

11k21533




11k21533










asked Dec 18 '18 at 16:52









oren1oren1

737




737








  • 1




    $begingroup$
    You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
    $endgroup$
    – JavaMan
    Dec 18 '18 at 16:54








  • 1




    $begingroup$
    Related post. Also, see the Polya conjecture.
    $endgroup$
    – Arthur
    Dec 18 '18 at 16:55






  • 1




    $begingroup$
    Dirichlet's Theorem addresses this question.
    $endgroup$
    – lulu
    Dec 18 '18 at 16:55












  • $begingroup$
    Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
    $endgroup$
    – oren1
    Dec 18 '18 at 17:06








  • 1




    $begingroup$
    Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
    $endgroup$
    – Bruno Andrades
    Dec 18 '18 at 17:29
















  • 1




    $begingroup$
    You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
    $endgroup$
    – JavaMan
    Dec 18 '18 at 16:54








  • 1




    $begingroup$
    Related post. Also, see the Polya conjecture.
    $endgroup$
    – Arthur
    Dec 18 '18 at 16:55






  • 1




    $begingroup$
    Dirichlet's Theorem addresses this question.
    $endgroup$
    – lulu
    Dec 18 '18 at 16:55












  • $begingroup$
    Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
    $endgroup$
    – oren1
    Dec 18 '18 at 17:06








  • 1




    $begingroup$
    Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
    $endgroup$
    – Bruno Andrades
    Dec 18 '18 at 17:29










1




1




$begingroup$
You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
$endgroup$
– JavaMan
Dec 18 '18 at 16:54






$begingroup$
You're looking for the modular prime counting function. Specifically, you want an estimate for $pi{4,3}(x)$.
$endgroup$
– JavaMan
Dec 18 '18 at 16:54






1




1




$begingroup$
Related post. Also, see the Polya conjecture.
$endgroup$
– Arthur
Dec 18 '18 at 16:55




$begingroup$
Related post. Also, see the Polya conjecture.
$endgroup$
– Arthur
Dec 18 '18 at 16:55




1




1




$begingroup$
Dirichlet's Theorem addresses this question.
$endgroup$
– lulu
Dec 18 '18 at 16:55






$begingroup$
Dirichlet's Theorem addresses this question.
$endgroup$
– lulu
Dec 18 '18 at 16:55














$begingroup$
Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
$endgroup$
– oren1
Dec 18 '18 at 17:06






$begingroup$
Very useful links. Still I leave the question open for I could not find bounds yet. I will edit my question again after researching these articles. Thanks!
$endgroup$
– oren1
Dec 18 '18 at 17:06






1




1




$begingroup$
Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
$endgroup$
– Bruno Andrades
Dec 18 '18 at 17:29






$begingroup$
Use the fact that for some fixed $N$ and an $a$ such that $(N,a)=1$ we have $$sum_{{pleq x}wedge{pequiv a mod N}}frac{ log p}p= frac{log x}{varphi(N)}+O_N(1)$$ this is Dirichlet's theorem
$endgroup$
– Bruno Andrades
Dec 18 '18 at 17:29












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