# Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution in number theory

106 questions
Filter by
Sorted by
Tagged with
30 views

• 1
1 vote
37 views

### Necessary and sufficient condition to be completely multiplicative

I want to prove that $f*f=f \tau$ iff $f$ is completely multiplicative. The "if" part was relatively easy, using $f(g*h)=(fg)*(fh)$ and plug $g=h=1$ for all $n$. Juxtaposition is ordinary, ...
• 140
75 views

### Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.

Here $f(n)=\begin{cases} 1\ \text{if } n=m^2\text{ for some }m\in\Bbb{N}\\ 0\ \text{if otherwise} \end{cases}$ This is a multiplicative function. At first I define $g:\Bbb{N}\to\Bbb{C}$ be $g(n)$ to ...
• 2,365
62 views

### direct computation of inverse of Dirichlet convolution

For an arithmetical function $f(n)$ with $f(1)\neq 0$ we have the recursive formula for the inverse (under Dirichlet convolution) of $f$ : $$f^{-1}(n)=(-1/f(1))\sum_{d|n, d<n}f(d)f^{-1}(n/d)$$ such ...
• 21
1 vote
59 views

• 770
47 views

### How to find the coefficent of a term in a Dirichlet generating function in Mathematica?

For a normal Dirichlet generating function like $Zeta[s]^2$, I can get the coefficient of the n-th term by applying Dirichlet convolution of the two constant functions. But how to find the coefficient ...
• 243
1 vote
45 views

82 views

Let be $\lambda: \mathbb{N} \rightarrow \mathbb{C}$ be an arithmetic function $$\lambda (n) := (-1)^{e_1+\dots+e_r}$$ where $p_1^{e_1}...p_r^{e_r}$ is the prime factorization of $n$ and it is $\... 1 vote 3 answers 154 views ### How does one obtain an expression for the Dirichlet series$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 ... • 5,017 1 vote 1 answer 77 views ### 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. • 561 2 votes 1 answer 64 views ### 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 ... • 561 1 vote 2 answers 144 views ### 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 : ... • 691 3 votes 3 answers 299 views ### Inversion theorem for Dirichlet series Can someone come up with a proof for this little theorem? Suppose that$F_a(s)$is a Dirichlet series and$a(n)$is its associated arithmetic function, that is: $$F_a(s)=\sum_{n=1}^{\infty}\frac{a(n)... 1 vote 0 answers 32 views ### Show that \sigma=c_1*c_1, where c_1 is the constant function 1. I searched and wasn't able to find a question similar enough to mine. Here's the problem: Show that \sigma=c_1*c_1, where c_1 is the constant function 1. Here is my attempt. My argument ... • 189 0 votes 1 answer 56 views ### Why can you replace 1/\sin(\phi/2) with 2/\phi in an integral? I am walking myself through a proof of convergence of Fourier series. For the partial sum S_Nf(\theta), and any constant S, we have that$$ S_Nf(\theta) - S = \frac{1}{2 \pi} \int_0^\pi ( f(\... • 121 -2 votes 1 answer 75 views ### Dirichlet-convolution Above is the definition i got from my note. I was trying to do these and i get stuck when i complete setting up the definition. I am trying to break down$c(n) = ((e_1 - 2e_2) * u)n$=$( u * e_1 - ...
• 173
1 vote
$p$ is prime Can someone show the intermediate steps, I don't understand the $1$st step even with the definition of a convolution in front of me. Thank you :)