Interested in the following function: $$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$ where $\pi(n)$ is the prime counting function. When $s=2$ the sum becomes the following: $$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...

Consider a random binary string where each bit can be set to 1 with probability $p$. Let $Z[x,y]$ denote the number of arrangements of a binary string of length $x$ and the $x$-th bit is set to 1. Moreover, $y$ bits are set 1 including the $x$-th bit and there are no runs of $k$ consecutive zer...

Consider an observer located at radius $r_o$ from a Schwarzschild black hole of radius $r_s$. The observer may be inside the event horizon ($r_o < r_s$). Suppose the observer receives a light ray from a direction which is at angle $\alpha$ with respect to the radial direction, which points outwa...

I've been told that the rational numbers from zero to one form a countable infinity, while the irrational ones form an uncountable infinity, which is in some sense "larger". But how could that be? There is always a rational between two irrationals, and always an irrational between two rationals, ...

$\mathbf{U} = (\mathbf{X}^H \mathbf{X} + \mathbf{I} + \mathbf{\lambda}\mathbf{F})^{-1} \mathbf{X}^HL $ $\rho(\lambda) = \mathbf{U}^H\mathbf{U}$. Find $\lambda \in (0,\infty)$ such that $$\frac{\partial{\rho}}{\partial{\lambda}} = 0$$ $X$ is $n\times m$ semi-orthogonal matrix $F$ is $m\times ...