The Prime Number Theorem relates primes to the important constant $e$.

Here I report my following surprising discovery which relates primes to $\pi$.

**Conjecture** (December 15, 2019). Let $s(n)$ be the sum of all primes $p\le n$ with $p\equiv1\pmod4$, and let $s_*(n)$ be the sum of those $x_py_p$ with $p\le n$, where $p$ is a prime congruent to $1$ modulo $4$, and $p=x_p^2+y_p^2$ with $x_p,y_p\in\{1,2,3,\ldots\}$ and $x_p\le y_p$. Then
$$\lim_{n\to+\infty}\frac{s(n)}{s_*(n)} = \pi.$$

Recall that a classical theorem of Euler (conjectured by Fermat) states that any prime $p\equiv1\pmod4$ can be written uniquely as $x^2 + y^2$ with $x,y\in\{1,2,3,\ldots\}$ and $x\le y$. Since $x^2 + y^2 \ge 2xy$ for any real numbers $x$ and $y$, we have $s(n) \ge 2s_*(n)$ for all $n=1,2,3,\ldots$.

I have created the sequence $(s_*(n))_{n>0}$ for OEIS (cf. http://oeis.org/A330487). Via computation I found that $$s(10^{10}) = 1110397615780409147,\ \ s_*(10^{10}) = 353452066546904620, $$ and $$ 3.14157907 < \frac{s(10^{10})}{s_*(10^{10})} < 3.14157908. $$ This looks an evidence to support the conjecture.

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

Any further check of the conjecture is welcome!

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