As claimed by Fermat and proved by Euler, any prime $p\equiv1\pmod4$ can be written uniquely as $s_p^2+t_p^2$ with $s_p,t_p\in\{1,2,3,\ldots\}$, $2\nmid s_p$ and $2\mid t_p$. For any positive integer $n$, let us define
$$S(n):=\sum_{p\le n\atop p\equiv1\pmod4}s_p
\ \ \ \text{and}\ \ \ T(n)=\sum_{p\le n\atop p\equiv1\pmod4}t_p.$$
Via computation, I found that
$$S(10^9)=334976550299,\ \ T(10^9)=334979004134,\ \ \frac{S(10^9)}{T(10^9)}\approx 0.99999267.$$
This leads me to pose the following conjecture.

**Conjecture.** We have
$$\lim_{n\to+\infty}\frac{S(n)}{T(n)}=1.$$

**QUESTION.** Is the conjecture true? If true, how to prove it?

Your comments are welcome!