Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution

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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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26 views

Bernstein Theorem on Banach Spaces

I've to do a presentation about a this Theorem, but during the study of it I found a lot of doubts, I will write it here down with the proof and doubts: Theorem: Let $P_n(z)$ be a trigonometric ...
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2answers
89 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)...
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0answers
19 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 ...
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0answers
32 views

Multiplications between four convolutions (discrete time).

Stuck on this, for long period. Need some help. Working on the thesis, PAM4 signal filtering. p - convolution noise x - input signal (PAM4) $$\ E(p^4[n]x^2[n]) = $$ Where E - is Expected Value and ...
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1answer
36 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(\...
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1answer
31 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 - ...
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58 views

Partial Differential Equations: Dirichlet problem for the Laplace equation on half-space with boundary condition

Consider the Dirichlet problem for the Laplace equation on half-space with the boundary condition $f(x)=\chi _{[0,\infty)}(x)$. Use the exact solution and find a functon $g:\mathbb{R}\rightarrow \...
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30 views

Partial Differential Equations: Consider the Dirichlet problem for Laplace equation on half-space with B.C, use exact solution, find a function g

i have no idea how to approach this problem! any help would be highly appreciated! thank you
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46 views

Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
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2answers
73 views

constant function under convolution with 3 *

$p$ 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 :)
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1answer
59 views

Easy way to prove property of prime indicator

Let $\mu$ be the Möbius function, and let $\nu(n)$ be the number of distinct prime factors of $n$. Then we can define $p = \mu * \nu$, i.e. $$ p(n) = \sum_{d \mid m} \mu(d) \nu(n/d). $$ An exercise in ...
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0answers
42 views

Square roots of the unity (DIrichlet convolution)

I am having a little trouble with this question. Given an arithmetic function f, a “Dirichlet square root” of f is an arithmetic function g such that $g ∗ g = f$. Prove by elementary techniques that ...
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1answer
119 views

Expression for the inverse of Euler's totient function $\phi^{-1}$

I have to demonstrate that $$\phi^{-1}(n)= \prod_{p|n}(1-p)$$ where $\phi(n)$ is the Euler's totient function. I know that I can write $\phi$ in terms of the Mobius function $\mu$ as$$\phi(n)= \sum_{...
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1answer
58 views

Let $f(x)$ be defined for all rational $x$ in $0\leq x\leq 1$

Let $f(x)$ be defined for all rational $x$ in $0\leq x\leq 1$ $$F(n)=\sum_{k=1}^n f\bigg(\frac kn\bigg), \quad F^* (n)=\sum_{k=1\\(k,n)=1}^nf\bigg(\frac kn\bigg).$$ Prove that $$F*=\mu * F$$ where $*$ ...
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33 views

How to invert an arithmetic function where Möbius inversion may not apply?

I'm playing around with problems in order to gain a very basic insight into number theory, and I am looking at the process of inversion. Take a convergent arithmetic function of the general form $$f(...
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1answer
32 views

Trinary Dirichlet convolution: $\sum_{abc=n} f(a)g(b) h(c)$ does not lead to anything new?

Defining $*(f,g,h)(n) = \sum_{abc=n} f(a)g(b)h(c)$ for arithmetic functions $f, g, h$. We have for instance: $*(f,g,h)(3) = $ " $(1,1,3) + (1,3,1) + (3,1,1)$ " where the tripple means the obvious ...
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1answer
57 views

Dirichlet convolution k times.

We know that $-\sum\limits_{d|n}\mu(d)\log d=\Lambda(n)$. Using this we can obtain $$(\Lambda*\Lambda)(n)=\Lambda(n)\log n+\sum\limits_{d|n}\mu(d)\log^2d.$$ In general if I write Dirichlet convolution ...
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1answer
141 views

Dirichlet Convolution [closed]

I was wondering how to simplify the expression: $id \ast (\mu\phi)$, where $\ast$ denotes Dirichlet convolution, $\mu$ is the Mobius function, $\phi$ is the Euler's totient function and $id$ is the ...
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2answers
231 views

Intuition about Dirichlet Kernel

I am studying Fourier analysis from the text "Stein and Shakarchi" and there is this thing on Dirichlet Kernel. It's fine to define it as a trigonometric poylnomial of degree $n$ , but what is the ...
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49 views

Proving that a set of functions with the Dirichlet convolution operation is a group

I want to prove that the set $S:(f:\Bbb Z\to\Bbb R);f(1)\neq 0)$ with the Dirichlet Convolution operator is a group. My process: $f,g:\Bbb Z\to\Bbb R$ and $f*g:\Bbb Z\to\Bbb R$ so $f*g\in S$ $f*(g*h)...
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1answer
151 views

Proof of $(\mu*1)=\delta_1$ where $(f*g)$ is Dirichlet Convolution

I was interested in the proof for this fact because it is used to prove M$\ddot o$bius Inversion Formula. However, I did not completely understand how proof wiki used a series of binomial coefficients ...
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69 views

On arithmetic functions whose Dirichlet series has a special kind of abscissa of absolute convergence

For a function $f: \mathbb N \to \mathbb C$ , let $\sigma_c(f) , \sigma_a(f)$ denote the abscissa of convergence and the abscissa of absolute convergence respectively of the Dirichlet series $\sum_{n=...
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3answers
73 views

Mysterious multiplicative function?

Consider $$f(n)=\prod_{p^k||n} p^{2k}(1+p^{-2})$$ Can this function be expressed by usual ones, as convolutions or directly? I do not know very well if convolution can be seen on decomposition in ...
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1answer
97 views

Properties of Dirichlet Inverse Series.

Let $F(n) = \sum f(n)n^{-s}$ that converges absolutely for $\sigma > \sigma_a$, where $f(1) \not= 0$. We may define the Dirichlet series, $G(s) := \sum f^{-1}(n)n^{-s}$, where $f^{-1}$ is the ...
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1answer
295 views

Theorem 11.14, Apostol, pg 238 - need explanation

An excerpt from Introduction to Analytic Number Theory by Tom M. Apostol. I have three main concerns regarding this proof: What is the abscissa of convergence for $\frac{1}{F(s)}$ - why can we take ...
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1answer
87 views

Convolution product for $2^{\Omega(n)}$

How can I write the multiplicative function $2^{\Omega(n)}$ as a Dirichlet product of two multiplicative functions? That's because I have to find an estimate for $\sum_{n\leq x}2^{\Omega(n)}$.
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55 views

Dirichlet Product - identities

Does anyone know of a good resource that shows how the number theoretic functions and their Dirichlet Products are related? This is for further reading but might come in useful for my exams. I am ...
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203 views

Is there a better convolution method for deriving $\sum_{p\le x}\frac{1}{p}$ when $p$ is an almost prime?

It's easy enough to derive an infinite sum for the logarithmic integral using the integral derived by Gauss through stepwise integration. For example, in my review of calculus I found: $$ li(x) - li(...
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2answers
494 views

Dirichlet Convolution of the Mobius Function with Itself

I am attempting to find a formula for $$(\mu * \mu)(n)$$ where * represents the Dirichlet Convolution operator. I know this can be expressed as $$\sum_{d|n} \mu(d)\mu(\frac{n}{d})$$ but I'd like the ...
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0answers
512 views

How to convolve a periodic signal with an aperiodic signal?

Basically, when there are two periodic signals, say x(t) and h(t) which are to be convolved, then convolution is carried out over a range of their common time period (which is equal to the least ...
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195 views

Dirichlet product of arithmetic functions $\tau$ and $\sigma$

Let $\tau$ and $\sigma$ be the arithmetic functions defined by $\tau(n)$ is the number of divisors of n, i.e $\tau(n) = \sum_{d\mid n}1$, $\sigma(n)$ is the sum of divisors of n , i.e $\sigma (n) = \...
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2answers
108 views

If $a$ is the arithmetic function with $\sum_{d\mid n}a(d)=2^n$, then $n\mid a(n)$

Problem: Let $a:\mathbb N \rightarrow \mathbb C $ a function with the property: $$\displaystyle{\sum_{d\mid n}a(d)=2^n, \forall n \in \mathbb N}.$$ Prove that $n\mid a(n), \forall n \in \mathbb ...
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2answers
113 views

Closed form for $\sum\limits_{d|n}\sigma(d)$?

Is there any closed form for $$\sum\limits_{d|n}\sigma(d)$$? I knew that $\sum\limits_{d|n}1=\sigma(n)$ therefore we must have $\sum\limits_{d|n}\sigma(d)=\sum\limits_{d|n}\sum\limits_{r|d}1$ but how ...
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2answers
151 views

Combinatorial proof of a relation between the number of distinct prime divisors of a number and the Riemann zeta function.

Recently I've been learning about the basics of Dirichlet functions and their relations to subjects in number theory that I would characterise as combinatorial in nature. I came across this identity: $...
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1answer
110 views

Does the dirichlet inverse of a series with finite abscissa of convergence also has a finite abscissa of convergence?

I would like to know if the Dirichlet inverse $L(s,g)$ of a series $L(s,f)$ ($f(1)\neq 0$) with finite abscissa of convergence also has a finite abscissa of convergence? Is there a specific criteria ...
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1answer
417 views

Dirichlet hyperbola methods : estimate functions # of ordered pairs

Let $f$ be an arithmetic function defined by $$f(n) = |A_n|$$ where $A_n = \{(a, b) : n = ab^2\}$.Estimate $$\sum_{n \leq x} f(n)$$ where $x \in \mathbb{R}^+$, using Dirichlet hyperbola method. The ...
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2answers
239 views

Compute the Dirichlet inverse of $f(n)=\frac{1}{1+|\mu(n)|}$, where $\mu(n)$ is the Möbius function

Let for integers $n\geq 1$ the arithmetical function defined by $$f(n)=\frac{1}{1+|\mu(n)|},$$ where $\mu(n)$ is the Möbius function. Note that $f(1)=\frac{1}{2}\neq 0$, and $f(n)$ isn't ...
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0answers
318 views

Question about Dirichlet kernel of Fourier transform for $f\in L^p$ with $p\in [1,2]$, help needed in understanding proof.

I am trying to understand the proof that the following two statements are equivalent. For fixed $R>0$ and $f\in L^p(\mathbb{R}^n)$ let $$S_Rf(x)=\int_{|\xi|<R} \hat{f}(\xi)e^{2\pi i x . \xi}\,d\...
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192 views

Associativity of the Dirichlet Convolution Product

How can you prove that the convolution product of aritmetical functions is associative, and that it is distributive in respect to the addition? The book that i'm reading states that (F_a, ) is a ...
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0answers
128 views

Calculating a multiple convolution with variables bounded both individualy and by total

I am trying to find a closed form or a transformation which simplify the numerical treatment of this multiple integral $$\int_0^{U_1} \cdots \int_0^{U_N} \delta(U,\sum_g u_g) \prod_g u_g^{n_g} \, du_1 ...
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0answers
55 views

What do we know about the set of Primitive Dirichlet Characters modulu an integer $N$?

I was working on Dirichlet's theorem on arithmetic progressions and I have the following question: Is the set of all Primitive Dirichlet Characters modulus some given (and fixed ) integer $N$ a ...
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1answer
170 views

Closed form or simplification of a multiple definit integral of a product of a weight averaged parameters

I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} {...
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2answers
154 views

Dirichlet Convolution of Mobius function and distinct prime factor counter function.

Let us define an Arithmetical function $\nu(1)=0$. For $n > 1$, let $\nu(n)$ be the number of distinct prime factors of $n$. I need to prove $\mu * \nu (n)$ is always 0 or 1. According to my ...
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1answer
127 views

On a generalization for $\sum_{d|n}rad(d)\phi(\frac{n}{d})$ and related questions

Let $\phi(m)$ Euler's totient function and $rad(m)$ the multiplicative function defined by $rad(1)=1$ and, for integers $m>1$ by $rad(m)=\prod_{p\mid m}p$ the product of distinct primes dividing $...
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1answer
43 views

Dirichlet, Möbius and Primes

Is it correct that for the Möbius function, for $p$ prime and $m>1$, $μ(p^m)= 0$ because there are repeated prime factors? So how would I use this if I have $f = μ ∗ μ$ to find $f(p^m)$ using ...
1
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1answer
163 views

Using Dirichlet convolution where f = μ ∗ μ (Mobius) to find f(24)?

I am confused about the Dirichlet convolution and how it is used. Does it take two entirely different arithmetic functions? And knowing that f = μ ∗ μ (the Mobius function), why does the question I ...
4
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1answer
358 views

Divisor sum of totient function

Is there any closed form expression for $\displaystyle\sum_{d|n} d\phi(d)$? I have tried a lot but can only reduce it to $\displaystyle\sum_{k=1}^{n}\frac{n}{(k,n)}$ where $(k,n)$ is the greatest ...
1
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1answer
152 views

Proof of the binomial theorem through Dirichlet convolution?

Here I gave a proof for $\sum_{k=0}^n\binom nk(-1)^k=0$ based on the fact that $\mu*1=\varepsilon$ (the Dirichlet convolution identity). I am wondering if using a similar technique we can prove that $\...
1
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1answer
476 views

Alternative proofs that Dirichlet products are associative?

Is there alternative proof of the following fact: Dirichlet product on arithmetic function is associative. I'm looking for something different than that given in Dirichlet's product with ...