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# Questions tagged [multiplicative-function]

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

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### If $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should be written for $\lambda$

Respected All. I am studying number theory where I came to know that $\varphi(n), \sigma(n)$ both are multiplicative function ; In other words, if $(m,n)=1$ then \begin{align} \sigma(mn)=\sigma(m)\...
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### How can I find The Multiplicative Inverse of $1+\sqrt{2}$? [closed]

I am doing contemporary abstract algebra and am working in an integral domain. I have found it necessary to compute the multiplicative inverse of $1+\sqrt{2}$; I know such the definition of a ...
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### Number theory proof for why $\tau$, the number of divisors of $n$ is multiplicative [closed]

Could someone please help me with a proof as to why $\tau$ is multiplicative?
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### $a\cdot(b^{-1}\bmod m)$ Can be be solved using modular multiplication

Does $a\cdot(b^{-1}\bmod m) = (a\bmod m) \cdot(b^{-1}\bmod m).$ where $\bmod$ represents remainder left on division with $m$. $b^{-1} \bmod m$ is multiplicative inverse.
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### What is known about these arithmetical functions?

Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and $$\alpha_N(n)=\prod_p p^{c_p \bmod N}.$$ The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
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### Writing product $\prod_{i=1}^m (p_i^{n_i}-1)$ as a sum

Suppose I have an integer $N$ with prime decomposition $N=\prod_{i=1}^m p_i^{n_i}$. How can I write $$\prod_{i=1}^m (p_i^{n_i}-1)$$ as a sum that only depends on $N$, and not it's prime ...
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### Proof to a property of Euler's totient function

The property is $$\sum_{d|n}\phi(d) = n$$ And the proof provided is If $d$ divides $n$, let $C_d$ be the unique subgroup of $\mathbb{Z}/n\mathbb{Z}$ of order $d$, and let $\Phi_d$ be the set of ...
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### Proving that a summation is multiplicative

I have been give a project for number theory: For $m>0$ , let $f(m) = \sum_{r=1}^m \frac{m}{\gcd(m,r)}$ . Evaluate $f(m)$ in terms of the prime factorization of $m$. So far, I have found a formula ...
859 views

### 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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### If a function $f$ is multiplicative, how do I show that $\sum_{d\mid n} \mu(n/d) f(d)$ is also multiplicative?

I am studying A Classical Introduction to Modern Number Theory by Ireland and Rosen, and this is exercise 9 from chapter 2. The authors define a function $f$ to be multiplicative if for all $a, b$ ...
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### formula for the number of perfect squares mod $N$

In a numerical experiment I notice for sum moduli $N$ there are much less than $N/2$ perfect squares. I had chosen a large number, the simplest example is $N=8$. Using the Chinese Remainder Theorem (...
344 views

### Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function. Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct ...
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### new addition and new multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z, prove the set Z equipped with these 2 new operation

here says a new operation addition and new operation multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z,where the operations on the right hand side are ordinary addition and multiplication ...
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### Euler-Totient Multiplicative

http://www.oxfordmathcenter.com/drupal7/node/172 By and large, I understand this proof, however I'm struggling to understand how the Chinese remainder theorem implies that there exists some $x \in S_1$...