Tagged Questions

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Is crossdirectional partial tetration of order $n^c$?

The following animation shows a sum of a matrix where the parameter $c=0$ gives a straight line in pink, and as $c \rightarrow 1$ it approaches the Chebyshev $\psi$ function, the blue staircase. This ...
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Proofs for $0^0 =1$? [duplicate]

Everyone knows the following: $$0^x = 0 \quad \wedge \quad x^0 = 1 , \quad\forall x \in R^*$$ One morning, I wake up asking myself the question "$\text{What is$0^0$, then?}$". So, I did what any ...
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Is $y(x)=\frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$ an integer

Let ${\rm y}\left(x\right) = \frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$. Is ${\rm y}\left(x\right)$ an integer for each $x = 1,2,\ldots,M$ when $M\to\infty$ ?.
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Limit of $\prod_{p\text{ prime}}\left(1-[p(p-1)]^{-1}\right)$?

Do we know the limit of the product $$\prod_{p\text{ prime}}\left(1-\frac{1}{p(p-1)}\right)$$ ? I ask because it seems to me on heuristic grounds (but I believe I could make them rigorous) that ...
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Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system. For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$. Question 1 :Is the following true? ...
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Is $\lim_{x\rightarrow\infty}\prod_{p\leqslant x}p^{1/x}$ finite?

I suspect $$\prod_{p \leqslant x}p^{1/(p-1)}\sim\sqrt[x-1]{x!}$$ and I was able to show that it is true if the finiteness of the above limit is true: ...
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REVISTED$^1$ - Order: Modular Arithmetic

Relevant Literature: Question: Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$. Thoughts: Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
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Probability of two random n-digit numbers dividing each other

Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
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How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$

Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$
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Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$

Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\displaystyle \lim_{n\to \infty}\frac{f(n)} n$. I suspect it's equal to $0$, but how can I show this? Thank you.
$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$ What does this limit indicate?