Questions tagged [multiplicative-function]
This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.
280 questions
0 votes
0 answers
26 views
Mutliplicativity of functions [duplicate]
So I am following a course on Elementary Number Theory right now and I have to prove the following if and only if statement: $$f \;\text{is multiplicative} \Longleftrightarrow g(n)=\sum_{d\mid n} f(d) ...
- 11
8 votes
1 answer
236 views
Question regarding the number of functions on $\mathbb{Z}/n\mathbb{Z}$ that satisfy properties similar to leibniz rule
This was a problem I came up with a while ago. The idea is to consider functions $f:\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ that satisfy $f(xy)=xf(y)+yf(x)$ for all $x,y\in\mathbb{Z}/n\mathbb{...
- 5,375
3 votes
2 answers
257 views
Asymptotic expansion of $\sum_{n \leq x} \frac{\mu^2(n)}{\phi(n)}$
How can we derive the asymptotic expansion $$ \sum_{n \leq x} \frac{\mu^2(n)}{\phi(n)} = \log x + C_0 + \sum_p \frac{\log p}{p(p-1)} + O(x^{-1/2} \log x)\,? $$ This question has been previously asked ...
- 1,058
0 votes
2 answers
206 views
Why is "the number of a rings of order $n$" a multiplicative function?
The OEIS sequence A037234 (also A027623) counts the number of (not necessarily unital) rings of order $n$. On the OEIS page (as pointed out by @Smiley1000) it says that the sequence is multiplicative. ...
- 2,386
5 votes
1 answer
195 views
How to correctly count "true" solutions to $x^n = 1 \bmod m$?
I got sidetracked (again) during the investigation of one of my other questions and started wondering how the cycles in the following picture came to be: What you see here is the graph generated by ...
- 1,489
3 votes
1 answer
119 views
The "Euler Product formula" for general multiplicative functions
For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$ This is easy to show ...
0 votes
1 answer
137 views
Show that the function $f(n) = \sigma(n)\varphi(n)$ is a multiplicative function and calculate $f(p^k)$. [closed]
Let $f(n) = \sigma(n)\varphi(n)$ where $\sigma(n)=\sigma_1(n)= \sum_{d|n}d$ and $\varphi(n)$ is Euler function. Show that $f(n)$ is a multiplicative function and calculate $f(p^k)$, where $p$ is a ...
- 39
14 votes
2 answers
584 views
Counting ones in binary representation: When is the product multiplicative?
Question: For $n \in \mathbb{Z}^+$, define $Z(n)$ to be the number of ones in the binary expression of $n$. For fixed positive integer $a$, how does one describe the set of $b$ such that $Z(ab) = Z(a)...
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