All Questions

Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...

What is $$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$ where $\mathbb N:=\{1,2,\dots\}$? In other words, I would like to ...

Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...

I would like to ask about a certain inequality that I need and which came out of some work in here. Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof? $$6\prod_{j=1}^k(...

QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that $$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\ 1-(1-(\frac{\...

I have already posted this question time before on stackexchange, but didn't receive a definitive solution. So this is the game: consider a positive integer number $n$ and divide it in a finite ...