Distribution of primes of 4n+1 and 4n+3 forms Can we give a $k$ such that the number of primes of the form $4n+1$ (upto m) will always exceed primes of the form $4n+3$ $\forall m>k$ or vice versa... $4n+3$ will always exceed primes of the form $4n+1$. Basically what I want is a stage where number of primes of one form outnumbers number of primes of the other form and it never retreats from there?
 A: No.
Search for
"littlewood primes of form 4n+1"
and you will find this article (https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf),
from which the following
is extracted:
It is a famous
(for a certain definition of famous)
result of Littlewood
that
$$\pi(x, 4, 3)-\pi(x, 4, 1)
=\Omega_{\pm}\left(\dfrac{x^{1/2}}{\log x}\log\log\log x\right)
$$
where $\pi(x, a, b)$ is the number of primes $\le x$ of the form $an+b$.
Meaning, the difference gets larger than a constant times the expression arbitrarily often in both positive and negative directions.
However,
if you use
logarithmic density,
and assume the Extended Riemann Hypothesis,
primes of the form 4n+3
occur more often
99.5% of the time.
It's a neat paper -
I learned a lot from it.
A: 
If the Riemann hypothesis for $\beta(s)=\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$ fails then $\pi_\beta(x) = \sum_{3 \le p \le x} (-1)^{(p-1)/2}$ changes of sign infinitely often.

Proof : if $\pi_\beta$ doesn't change of sign for $x> A$ then $$\log \beta(s) = g(s)+s\int_A^\infty \pi_\beta(x)x^{-s-1}dx$$ where


*

*$g$ is analytic for $\Re(s) > 1/2$ 

*$\int_A^\infty \pi_\beta(x)x^{-s-1}dx$ has a singularity at $s = \sigma$ its abscissa of convergence

*$\sigma> 1/2$ since the RH fails for $\beta$
We know it isn't the case since $\beta(s)$ doesn't vanish on $(0,\infty)$ thus $\log \beta(s)$ is analytic there.
A: In 1853, Chebyshev indicated that he had a proof that the number of primes of the form $4n+1$ is less than the number of primes of the form $4k+3$. However, in 1914, Littlewood showed that Chebychev's assertion fails infinitely often. Therefore, there does not exist a prime $p$ of type $4n+1$ or $4n+3$ beyond which the aggregate of primes of that type remains the leader in what has been termed the ``prime number race.''
