Special case of prime number theorem for arithmetic progressions 4k+1

Question

In terms of the proof of prime number theorem for arithmetic progressions, I have seen many proofs involving with the concept of "character". Is there an alternative way (without such a concept) to show that the number of primes which are congruent to 1 modulo 4 is asymptotic to $\dfrac{x}{2\log x}$?

Answer 1

To prove that

$$ \frac{\#(\text{primes}~~ qn+a \leq x)}{\pi(x)/\phi(q)}\to 1~ \text{as} ~ x\to \infty, $$

that there are roughly the same number of primes in each residue class $a$ if $(a,q)=1$ without some version of Dirichlet characters/series might be hard.

There is no special virtue of using $Li(x)$ here, any function $\sim \pi(x)$ will work.

Hadamard and de La Vallée Poussin are credited with the first proof of this. The most economical proof I can find is that in G.J.O. Jameson's The Prime Number Theorem at pages 175-176. It depends on the sorts of calculations you specifically ask to avoid.

In light of comments it seems like a tall order.

Edit: Regarding the hint in the comment about points on the circle $a^2+b^2=c^2$ I remembered some theorems that are in the linked paper. The main idea is that for primitive Pythagorean triples (a,b,c), c is always of the form $4n+1.$ Assuming all primes of this form occur in the sequence of primitive triples I do not see how to show that their cardinality is $\sim$ the cardinality of primes $4n+3.$ If the idea is showing that the cardinality of primes c is 1/4 of numbers generally I don't see that either.