# 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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### Prove of $\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$

Found an interesting relation: $$\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$$ where $\mu(n)$ is a Möbius function and $d(n)$ is a divisors count. I think this should be something known. The prove ...
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### How does this multiplication work?

I'm studying the CL signature that how can it works. However, I cannot understand how equation (42) is calculated. What is the last three factors? The value T is as follows: The full paper link is ...
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### Reference request: Axiomatic treatment of multiplicative functions?

I'm currently reading Apostol's analytic number theory, Chapter 2 on multiplicative functions. While the current exposition is nice, I can't help but feel that there has to been some algebraic ...
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### Name of Möbius type operator?

Does anyone know if the Möbius / Dirichlet - type operator: $$f \mapsto \sum_{d|n} f(d)$$ that (among other things) appear in the Möbius inversion formula has a name? Would it be fair to call it the ...
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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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### 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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### 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 example, ...
$f(n)$ 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 ...