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
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What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel ...
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Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\$ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (link to ...
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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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On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x)$

There is a second part of the problem posted in Proving $\frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$, from Apostol's book, but I can't figure it out. It ...
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Series of the totient function

Good morning, I wonder if : $$\sum_{n} \frac{(-1)^n}{\varphi (n)}$$ converges or not. where $\varphi (n)$ is the Euler function. Do you have any idea ?
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Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
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Some convincing reasoning to show that to prove that Ramanujan tau function is multiplicative is very difficult

I am curious to know (some words or reasoning about how to justify) why to prove that the so-called Ramanujan $\tau$ is a multiplicative function is (was) very difficult. In this Wikipedia is showed ...
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Is there a recursive formula for Euler's Totient function

I have been looking for a recursive formula for Euler's totient function or Möbius' mu function to use these relations and try to create a generating function for these arithmetic functions.
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$\tau(n)\phi(n)\ge n$

how to prove $\forall n \in \Bbb N$ $$\tau(n)\phi(n)\ge n$$ $\tau(n)$ is number of positive divisor of $n$ my efford: if $n=p$ is prime then $\tau(p)=2,\phi(p)=p-1,2p-2\ge p$ but how prove for ...
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Can $\sigma(p^k)=2^n$ for some $k>1$?

If $k=1$ then the only solutions to $$\sigma(p^k)=2^n$$ are when $p$ is a Mersenne prime (of course $p$ is restricted to the primes). Is there a solution for larger $k$? It doesn't seem so but I can'...
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Prove $\sum_{k\mid n}{\mu(k)d(k)}=(-1)^{\omega{(n)}}$

I have the following exercise. Show that for all natural numbers $n$, the following equality holds $$\sum_{d|n}{\mu{(d)}d(d)}=(-1)^{\omega{(n)}}$$ Here, $\mu$ is the Möbius function, $d$ ...
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Is $\sum_{d | n} f(d)$ completely multiplicative if $f$ is completely multiplicative?
In these excellent notes by Pete L. Clark, it is claimed that if $f(n)$ is a multiplicative function, then $F(n) = \sum_{d|n} f(d)$ is multiplicative as well. I understand the proof of this statement,...