# Tagged Questions

53 views

### How to show that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ [duplicate]

set $n, n \in \mathbb{N}$ and prove that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ ¨I have tried this¨ If $n > 1$ then $n = p_{1}^{\alpha_{1}}\cdot p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$ ...
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### sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

Does there exist an asymptotic estimate for the following sum over primes $$\sum_{p\leq x} \frac{\tau(p-1)}{p}\;,$$ where $\tau(n)=\sum_{d|n}1$ is the divisor function?
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### How to show that $f(p^{k}) = f(p) \cdot f(p^{k-1}) \Longrightarrow f(p^{k}) = [f(p)]^{k}$ [closed]

If f is an arithmetic function such that $f (1) = 1$ and $p$ is a prime number. Prove that: $\forall k \in \mathbb{N}$ $f(p^{k}) = f(p) \cdot f(p^{k-1}) \Longrightarrow f(p^{k}) = [f(p)]^{k}$
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### Is the totient function $\varphi$ invertible?

As title, is the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ invertible?
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### Prove that the Möbius function is multiplicative

I'm studying algebra, and I came across some questions on multiplicative functions (that should be number theory though?). One is: prove that mobius function is multiplicative. But I've not been given ...
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### On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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### A Möbius Identity

I need to prove the identity $$\sum_{k = 1}^n \mu(k)\left[ \frac{n}{k} \right] = 1$$ where $n$ is a natural number, and $[n]$ denotes the floor function. The proof also should not use the Möbius ...
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### prove that $\phi(xy) =\phi(x)\phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. [duplicate]

Prove that $\phi(xy) = \phi(x) \phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. I understand the concept, and have done several examples proofing this but cannot put it in "proof form" because unless ...
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### An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
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### Arithmetic progressions that generate an infinite number of powers of 2?

For an arithmetic progression of the form $a_i = ki, i \in \mathbb{N}$, the question is trivial - if $k$ is a power of 2, then the progression will generate an infinite number of powers of 2, and no ...
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### For every positive integer n greater than $2$, $\phi(n)$ is an even integer.

Theorem: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer. I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show ...
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### If $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$

Show that if $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$ I tried, $(m,n)=1$ then, Euler theorem we have $m^{\phi (n)}\equiv1\pmod n$ and $n^{\phi (m)}\equiv1\pmod m$. But I could ...
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### Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$ I found (by trial) $m=\{13,21,26,28,36\}$, but do not know if misinterpreted the problem, but actually I suppose I have to find an equation ...
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### Prove that for all $n\in\mathbb{N}$, $\frac{s(n)}{d(n)}\geq \sqrt n$

Prove that for all $n\in\mathbb{N}$ $$\frac{s(n)}{d(n)}\geq \sqrt n$$ where $s(n) = \sum_{d|n} d$ and $d(n) = \sum_{d|n} 1$. Being honest, study some time arithmetic functions, and can not ...
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### Demonstration in arithmetic function

I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function. $f(n)$ and $g(n)$ are functions multiplicative, ...
### Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?
Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...