Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution

66 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...