# Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution

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### Zero divisors in the Dirichlet ring

I'm trying to determine if the ring $(\mathbb{A}, +, *)$ is an integral domain, where $\mathbb{A}$ is the set of arithmetic functions and $*$ is the Dirichlet convolution. To do so, I'm trying to ...
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### Dirichlet convolution of Mobius function with exponential function

Define $\exp_x : \mathbb{N} \rightarrow \mathbb{C}$ by $\exp_x(d) = e^{ixd}$ for all $d \in \mathbb{N}$ and some $x \in \mathbb{R}$. I want to evaluate the Dirichlet convolution of the Mobius function ...
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### Internal binary operation

Dirichlet Convolution. If $f,g:\mathbb {N} \to \mathbb {C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution $f ∗ g$ is a new arithmetic ...
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### Regarding expressing Lambert series in terms of Dirichlet Convolution

I am studying Lambert Series . It's definition says a series of the form $\sum_{n=1}^\infty \frac { f(n) x^n } { 1 - x^n }$ = $\sum_{n=1}^\infty F(n) x^n$ , where $F(n) = \sum_{d|n} f(d)$ ....
1answer
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### Dirichlet Self-Convolution Inversion

I am interested in finding out a method to invert Dirichlet selfconvolution. In math expressions it means: Find out $a$ once $b=a*a$ is known So a kind of squareroot of the Dirichlet product. I ...
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### Alternative way to compute some sums from number theory.

Let $*$ stands for Dirichlet's convolution operator I know that sum of the form $$\sum_{1\le n\le x}(f*g)(n)$$ is somehow easy to compute/estimate using methods such: changing order of summation ...
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### How does one obtain an expression for the Dirichlet series $g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}}$?

I would like to obtain an expression for the function $$g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}} \qquad (\#).$$ Here is what I've tried so far: we know from the definition of ...
1answer
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### How to prove that $\sigma_{k+1}(n) = \sum\limits_{d|n}d^k \cdot\phi(d)\cdot\sigma_{k}(\frac{n}{d})$

How to prove that $\sigma_{k+1}(n) = \sum\limits_{d|n}d^k \cdot\phi(d)\cdot\sigma_{k}(\frac{n}{d})$ I've tried using Dirichlet's convolution but the $d^k$ term seems to be something I can't resolve.
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### Alternate method to derive the following equation

$$\sum\limits_{d\mid n} {\tau (d)\varphi (n/d) = \sigma (n)}$$ I have seen a derivation based on Dirichlet convolution: Relation between $\sigma (N)$, $\tau (N)$, and $\varphi (N)$ Is there another ...
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### Convolutions : find $f*g(x)=\int_{R}f(x-y)g(y)dy$

Today I need calculate this convolution : Given : $f(x)=e^{x}1_{]-a,a[}(x)$ and $g(x)=e^{-x}1_{]-b,b[}(x)$ Where : $a<b$ , $a,b\in R$ Then find : $f*g(x)=\int_{R}f(x-y)g(y)dy$ My try : ...
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