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.

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How to prove that the Legendre symbol is multiplicative?

The proof is given here in the answer Proving $(\frac{n}{p})$, a Legendre symbol, is multiplicative But I do not understand it, Also the definition in the book for Legendre symbol says that if $p|a$ ...
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Explain why the following is or isn't a multiplicative function.

I'm working through practice problems for my exam in class and I need some help with the following problem: A function $f(n)$ is defined to be the greatest power of $2$ that divides $n$. For ...
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Multiplicative functions and the sum of all divisors: $\sum_{d\mid2020}{\sigma(d)}$

Doing more practice for my final and I need some help with the following: Evaluate: $$\sum_{d\mid2020}{\sigma(d)}$$ where $\sigma(n)$ is the sum of all divisors of n. The hint given specifically ...
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Multiplicative function evaluated at prime decompositions

Say $f$ is multiplicative and $m = p^{\alpha}q^{\beta}$ where $p,q$ are prime. Then do we have $f(m) = f(p^{\alpha})f(q^{\beta})$? If so why?. I know this hold when the powers are 1 but have not been ...
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Summing a multiplicative function

$f(n)$ is a multiplicative function, meaning $f(m\cdot n)=f(m)\cdot f(n)$. I want to evaluate the sum: $$(1)\qquad\sum_{k=1}^{n}f(m\cdot k)$$ over a fixed $m$. Because $f$ is multiplicative, I can ...
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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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Is $n^{(p-1)/2}\equiv -1\pmod p$ implies $n$ is a primitive root modulo $p$?

Let $p$ be an odd prime, $n$ be any integer, if $$n^{(p-1)/2}\equiv -1\pmod p,$$ is it always true that $k=p-1$ is the smallest positive integer satisfy $$n^k\equiv 1\pmod p?$$ This is the little ...
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Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...
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If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.

I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
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I am trying to multiply and simplify the following radical expression. $$(\sqrt{x}+5 - 4)(\sqrt{x}+5+4)$$ According to the book, the answer is $x - 11$ However, I am confused about how this even ...
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