All Questions

Given a positive real (or integral) number $x$ we consider the electrostatic potential energy of equal point charges at all primes up to $x$ given by $$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$ ...

I have been playing with the following function: $$ f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k} $$ It is hard to get correct numerical values. I'll start with ...

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'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 ...