# Tagged Questions

In number theory, a multiplicative function is a function defined on positive integers such that f(ab)=f(a)f(b) for a,b coprime. E.g. Euler's totient function, sum of divisors and number of divisors are multiplicative functions.

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### A multiplicative property of the Euler totient function $\phi$ [duplicate]

How can I show that if $\gcd(a,b)=d$, then $$\phi(ab)= {\phi(a) \phi(b) d \over\phi(d)}$$ I know I have to use the fact that $$\phi(m)= m \cdot\prod_{p|m} (1-\frac1p)$$, where the $p$ ranges ...
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### Number theoretic function related to totient

I'm doing an excercise in Alan Baker's book A Concise Introduction to the Theory of Numbers, and I'm confused about the method spelled out for one question. I'll quote it here: Let $a$ run through ...
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### Prove that $f$ is a multiplicative function and calculate the Summatory Function

Define $f(n)$ as $1$ if $n$ is odd, and $3$ if $n$ is even. So i have $f(odd) = 1$ and $f(even) = 3$ If a function $f$ is multiplicative then if $gcd(m,n) = 1$ then $f(m * n) = f(m) * f(n)$ This ...
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### Why is this expression returning NaN? [closed]

For smoother ship movement, I am going to gradually move it to it's actual location - So like this: (mz - sz) / (10 * (60 / TerrainDemo.FPS)) mz is the ship's ...
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### Grade School Multiplication Algorithm for Binary Numbers explanation

I under stand the shifting but not why it will always give the right answer? For Example: ...
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### Prove $\lambda(n)=\sum_{d^2|n}\mu(n/d)^2$ and $\mu^2(n)=\sum_{d^2|n}\mu(d)$

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
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### Prove that $g(n)$ is a multiplicative function as well

Suppose that $f(n)$ is any multiplicative function , and define a new function as $g(n) = f(d_1) + f(d_2) +...+f(d_r)$, where $d_1, d_2,...,d_r$ are divisors of $n$. Prove that $g(n)$ is a ...
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### Very elementary proof of that Euler's totient function is multiplicative

Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function. My problem is this: I'd like to explain this to some ...
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### Summation of multiplicative function $f$ where $f(p) = 1$ for $p$ prime

I have a multiplicative function $f$ with a special "base" case: For every prime $p$, $f(p) = 1$. E.g. splitting up $f(3^5 \times7^2 \times 13 \times 17)$ yields $f(3^5) f(7^2)$ which is left to be ...