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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 ...

I am looking for the proof of the following claim: First, define the function $\operatorname{sgn_1}(n)$ as follows: $$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...

This is a repost of this question . Can you provide a proof or a counterexample to the claim given below ? First , we shall give a definition of the Inkeri's primality test for Fermat numbers : ...