Over the past few months, I have been studying a few number theory topics (my main area of interest is operator theory) which naturally led me to the near-square prime problem. I came up with the following question while I was reading articles/books on Hardy-Littlewood conjecture.
What is the limit of the two products involving prime numbers modulo 4? $$\lim_{x \to \infty}\frac{\prod_{p \leq x \\ p \equiv 1 \pmod{4}}\left(1-\frac{1}{p}\right)}{\prod_{p \leq x \\ p \equiv 3 \pmod{4}}\left(1-\frac{1}{p}\right)}$$
Any idea or reference will be greatly appreciated.
The Landau-Ramanujan constant has an expression $$ K = \frac{1}{\sqrt{2}}\prod_{p\equiv 3(4)}\left(1-\frac{1}{p^2}\right)^{-1/2}. $$ Your limit can be written in terms of this constant: $$ \begin{align*} \lim_{x\to\infty}\frac{\prod_{\substack{p\leq x\\p\equiv 1}}\left(1-\frac{1}{p}\right)\prod_{\substack{p\leq x\\p\equiv 3}}\left(1+\frac{1}{p}\right)}{\prod_{\substack{p\leq x\\p\equiv 3}}\left(1-\frac{1}{p}\right)\prod_{\substack{p\leq x\\p\equiv 3}}\left(1+\frac{1}{p}\right)}&=\frac{\prod_p \left(1-\frac{\chi(p)}{p}\right)}{\prod_{p\equiv 3}\left(1-\frac{1}{p^2}\right)}\\ &=2L(1,\chi)^{-1}K^2=\frac{8 K^2}{\pi}, \end{align*} $$ where $\chi$ is the non-trivial Dirichlet character mod 4.