I am looking for a proofs of the following two claims:
Claim 1.
$$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime factors of the form $p \equiv 1 \pmod{6}$ of $n$ .
The SageMath cell that demonstrates this claim can be found here.
Claim 2.
$$\frac{\sqrt{3}\pi}{2}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_5(n)}}{n}$$ where $\Omega_5(n)$ is the number of prime factors of the form $p \equiv 5 \pmod{6}$ of $n$ .
The SageMath cell that demonstrates this claim can be found here.
Best Answer
You recently asked a similar question for modulus $4$ on math.stackexchange. I just used the exact same technique.
For $i\in\{1,5\}$, define $f_i:\mathbb{Z}_{\ge 1}\to \mathbb{C}^\times$ by $f_i(n)=(-1)^{\Omega_i(n)}$, then $f_i$ is completely multiplicative and $$L(s,f_i)=\left(1-2^{-s}\right)^{-1}\left(1-3^{-s}\right)^{-1}\prod_{p\equiv i\pmod 6}\left(1+p^{-s}\right)^{-1}\prod_{p\equiv -i\pmod 6}\left(1-p^{-s}\right)^{-1}.$$ Hence, $$L(s,f_1)L(s,f_5)=\left(1-2^{-s}\right)^{-2}\left(1-3^{-s}\right)^{-2}\prod_{p\ge 5}\left(1-p^{-2s}\right)^{-1}=\frac{2^{s}+1}{2^s-1}\cdot\frac{3^s+1}{3^s-1}\cdot \zeta(2s).$$ Therefore, if both $L(1,f_1)$ and $L(1,f_5)$ converge, their product is $6\zeta(2)=\pi^2$. Now, let $\chi$ be the non-trivial Dirichlet character modulo $6$, then $$L(1,\chi)=\prod_{ p\equiv 1\pmod 5}(1-p^{-1})^{-1}\prod_{p\equiv 5\pmod{6}}\left(1+p^{-1}\right)^{-1}=\frac13L(1,f_5).$$ Therefore, it suffices to show that $L(1,\chi)=\frac{\pi}{2\sqrt{3}}.$
Let $\chi_2$ be the non-trivial Dirichlet character modulo $3$, then $\chi(p)=\chi_2(p)$ for all primes $p\neq 2$ and $$L(1,\chi)=\frac{L(1,\chi_2)}{(1+2^{-1})^{-1}}=\frac32 \cdot \frac{\pi}{3\sqrt 3}=\frac{\pi}{2\sqrt{3}}$$ Where we use this answer.