Questions tagged [multiplicative-function]

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.

189 questions
652 views

How to find all elements in Z/80 that have multiplicative inverses.

I need to find all the elements in Z/80 that have multiplicative inverses. Z/80 is not a field, so I know not every element will have an inverse. Is there a shorter way than just writing out the ...
139 views

Closed formula for a multiple Dirichlet convolution of the 1-function with the identity

For two multiplicative arithmetic functions $f,g$ the Dirichlet convolution is defined by $(f\ast g) (n)=\sum\limits_{ab=n}f(a)g(b)$. Convoluting any arithmetic function with the $1$-function ($1(n)=1$...
685 views

Why f(1)=1 for every multiplicative function f?

If $f$ is a multiplicative function with $f(1)\ne0$, then why is $f(1)$ necessarily equal to $1$?
218 views

Multiplicative inverses and co-primes

I'm working out some examples on multiplicative inverses. I understand how to solve for a multiplicative inverse using the Extended Euler's algorithm, but I don't understand the principles which ...
183 views

How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

show that $$\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$$ where $d(n)$ is the number of positive divisors of $n$. see this have simaler $$1^3+2^3+\cdots+n^3=\left(1+2+\cdots+n\right)^2$$ maybe ...
50 views

Summatory Function $F(n) = 1$ for all $n$ odd, and $F(n) = 2$ for all n even

So, I have this summatory function $$F(n)=\sum_{d\mid n}f(n)$$ that goes $F(n) = 1$ for $2\nmid n$, and $F(n)=2$ for $2\mid n$. This summatory function is multiplicative. I need to describe the ...
1k views

Prove that if d = gcd(m,n) then $\phi(mn)=\phi(m)*\phi(n)/d$ [duplicate]

So if m and n are relatively prime, then the $\phi(mn)=\phi(m)*\phi(n)$ but what happens when $d > 1$?