All Questions

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$. In a hunt for an "...

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'm trying to estimate the product $$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$ where $p,q,r,s$ are primes. This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...