Questions tagged [multiplicative-function]

This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.

In number theory, a multiplicative function is an arithmetic function $~f~ \colon \mathbb N \to \mathbb C$ fulfilling $f(1)=1~~,~~f(ab)=f(a)~f(b)~~$ for $~a~$, $~b~$ coprime.

A completely multiplicative function satisfies $~f(ab)=f(a)~f(b)~$ for all values of $~a~$ and $~b~$.

Multiplicative functions arise naturally in many contexts in number theory and algebra. The Dirichlet series associated with multiplicative functions have useful product formulas, such as the formula for the Riemann zeta function.

Well-known examples of multiplicative functions:

  • Euler's totient functions $\varphi(n)= \text{the number of positive integers $~a\le n~~$ such that} ~~gcd(a,n)=1~.$

  • Divisor functions, e.g. number of divisors $d(n)$ and sum of divisors $\sigma(n)$

  • The Möbius function: $~\mu(n)$

  • The function $e(n) = \left\lfloor\frac{1}{n}\right\rfloor == \begin{cases} ~1~ \quad \text{if}~~ n=1\\ 0 \quad \text{otherwise} \end{cases}$
  • The unit function: $~\mathbf{1}(n)=1~$
  • The identity function: $~I(n) = n ~$

References:

https://en.wikipedia.org/wiki/Multiplicative_function

https://brilliant.org/wiki/multiplicative-function/

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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 that its multiplicativity should be an approximation…
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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 Google books sample), the open problems and…
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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 gifted children. The children are gifted enough to…
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How to find the nearest multiple of 16 to my given number n

If I'm given any random $n$ number. What would the algorithm be to find the closest number (that is higher) and a multiple of 16. Example $55$ Closest number would be $64$ Because $16*4=64$ Not $16*3=48$ because its smaller than $55$.
Mrshll187
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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 asks the reader to prove that if $x \geq 2$ then…
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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 ?
user279923
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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 express the main term constant in terms of values of…
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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 the conjecture due to Ramanujan, and I see that…
user243301
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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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Why doesn't $255 \times 255 \times 255 = 16777215$

Ok, I obviously understand basic multiplication and understand why those don't equal. But in web colors, therr is FFFFFF hexadecimal different colors (or rather $16,777,215$ in base $10$). This amount of colors can also be described by using three…
tysonsmiths
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Let $(a_n)$ be a strictly increasing sequence of positive integers such that: $a_2 = 2$, $a_{mn} = a_m a_n$ for $m, n$ relatively prime.

Let $(a_n)$ be a strictly increasing sequence of positive integers such that: $a_2 = 2$ and $a_{mn} = a_m a_n$ for $m, n$ relatively prime. Show that $a_n = n$, for every positive integer $n$. This is a result apparently due to Paul Erdős, and…
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A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The proposition on hand is the following: If for…
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Counting which values of a polynomial in $\mathbb{Z}[X]$ are coprime to a given integer.

The following problem is from Ireland and Rosen's Intro to Modern Number Theory, Exercise 23 Chapter 2. Let $f(x)\in\mathbb{Z}[X]$ and let $\psi(n)$ be the number of $f(j)$, $j=1,2,\dots,n$ such that $(f(j),n)=1$. Show that $\psi(n)$ is…
yunone
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Minimum possible value of $f(2007)$ where $f(m f(n)) = n f(m)$, $m,n\in \Bbb N$

If $f$ is from positive integers to positive integers and satisfies $f(m f(n)) = n f(m)$ then find the minimum possible value of $f(2007)$. My work so far: $f(1) = 1$ . Proof: Suppose $f(1) = k \neq 1$. Then consider $f(f(2)) = 2f(1) = 2k$. …
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Prove that $f(n)=n^2$ where $f$ is a strictly increasing multiplicative function with $f(2)=4$.

Let $f:\mathbb N\to\mathbb N$ be a strictly increasing function with $f(2)=4$ which is completely multiplicative i.e $f(ab)=f(a)f(b)$ for all $a,b\in\mathbb N$. Prove that $f(n)=n^2$ for all $n\in\mathbb N$. This is an exercise on induction. So I…
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