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From: Mike on 13 May 2010 16:34
Does anyone know how to prove that there are "as many" primes
congruent to 1 as to 3 modulo 4? My technical meaning of "as many" is
this: Let A(n) be the number of primes < n that are = 1 mod 4 and let
B(n) be the number of primes <n = 3 mod 4. Is it true that A(n) /
B(n) approaches 1 as n goes to infinity? More generally, it is known
that if (a,b) = 1 then there are infinitely many primes congruent to b
mod a. This suggests:

Conjecture: If a,b, and c are integers such that (a,b) = 1 and (a,c)
= 1 then there are as many (in the above sense) primes congruent to b
mod a as to c mod a.

Does anybody know a reference for questions of this sort?
From: Raymond Manzoni on 13 May 2010 17:00
Mike a �crit :
> Does anyone know how to prove that there are "as many" primes
> congruent to 1 as to 3 modulo 4? My technical meaning of "as many" is
> this: Let A(n) be the number of primes < n that are = 1 mod 4 and let
> B(n) be the number of primes <n = 3 mod 4. Is it true that A(n) /
> B(n) approaches 1 as n goes to infinity? More generally, it is known
> that if (a,b) = 1 then there are infinitely many primes congruent to b
> mod a. This suggests:
>
> Conjecture: If a,b, and c are integers such that (a,b) = 1 and (a,c)
> = 1 then there are as many (in the above sense) primes congruent to b
> mod a as to c mod a.
>
> Does anybody know a reference for questions of this sort?


See for example here :
<http://en.wikipedia.org/wiki/Prime_number_theorem#The_prime_number_theorem_for_arithmetic_progressions>
and
<http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions>

Hoping this helped,
Raymond
From: José Carlos Santos on 14 May 2010 09:06
On 13-05-2010 21:34, Mike wrote:

> Does anyone know how to prove that there are "as many" primes
> congruent to 1 as to 3 modulo 4? My technical meaning of "as many" is
> this: Let A(n) be the number of primes< n that are = 1 mod 4 and let
> B(n) be the number of primes<n = 3 mod 4. Is it true that A(n) /
> B(n) approaches 1 as n goes to infinity?

I don't know, but Littlewood proved that the sign of A(n) - B(n) changes
infinitely often; see

J. E. Littlewood
Sur la distribution des nombres premiers
C. R. hebd. S�anc. Acad. Sci. Paris 158(1914), pp. 1868-1872

See also

W. W. L. Chen
On the Error Term of the Prime Number Theorem and the Difference
between the Number of Primes in the Residue Classes Modulo 4
J. London Math. Soc. 1981 s2-23: 24-40

Best regards,

Jose Carlos Santos
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