Questions tagged [dirichlet-convolution]

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0answers
36 views

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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47 views

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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27 views

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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0answers
18 views

Dirichlet convolution of a multiplicative function with itself

Let $f$ be a multiplicative function, i.e. for all coprime $a,b\in\mathbb{N}\quad f(ab)=f(a)f(b)$. Consider: $$(f*f)(n)=\sum_{d\vert n}f(d)f\left(\frac{n}{d}\right),\quad\text{where * is Dirichlet ...
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39 views

Dirichlet Energy for Graphs, Derivation

I would like to prove this formulation of the Dirichlet Energy for Graph Neural Networks $$ \begin{aligned} E(\mathbf{X}) &=\frac{1}{d_{i}} \sum_{j \in \mathcal{N}(i)} w_{i j}\left\|\mathbf{x}_{i}-...
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1answer
56 views

Why does the proof of Theorem 3.10 in Apostol (1976) use generalized convolutions from previous chapter when there could be a simpler proof?

The proof of Theorem 3.10 in Introduction to Analytic Number Theory by Apostol goes like this: Theorem 3.10 If $ h = f * g $, let $$ H(x) = \sum_{n \le x} h(n), F(x) = \sum_{n \le x} f(n), \text{ and ...
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64 views

Pulse train rect function

Why $ X_\delta (f) = \frac{4A}{3} ( 1 - \frac{1}{2} rect \frac{f}{B} ) $ is equal to The $ X_\delta (f) $ signal I wrote here ? my book obtained the same equation I wrote on the paper but , before ...
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51 views

Dirichlet Inverses of Binomial Coefficients

Let $\omega$ be a real number between $0$ and $1$, and let: $$\mathbf{c}\left(n\right)=\binom{\omega+n-1}{n}$$ for all positive integers $n$. Is there a closed form for the Dirichlet inverse $\mathbf{...
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70 views

Apply the method of hyperpolas to $\sum_{n \le {x}^{1/k}} n * Fraction \left(\frac{x}{{n}^{k}}\right)$

I am using the example from Theorem 1 of Friedrich Pillichshammer "Euler's Constant and Averages of Fractional Parts" (https://www.dmg.tuwien.ac.at/nfn/gamma.pdf) we have for integers $k >...
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2answers
67 views

Is it possible turn the Dirichlet ring into a Banach algebra?

The set of all arithmetic functions $f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible. So my ...
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1answer
60 views

Dirichlet convolution of the small prime omega function and the Mobius function

I have seen that: $$(\omega\star\mu)(n)=\sum_{d\vert n}\mu(d)\omega\left(\frac{n}{d}\right)=\begin{cases}1 & n\ \text{is prime}\\ 0 &\text{otherwise} \end{cases}$$ where $\mu(n)=\delta_{\omega(...
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26 views

Does $ \mu * N $ imply $ \mu^{-1} * N^{-1} $ where $ * $ denotes Dirichlet multiplication?

Quoting from the book Introduction to Analytic Number Theory by Tom A. Apostol > Section 2.11 (page 37). EXAMPLE The inverse of Euler's $ \varphi $ function. Since $ \varphi = \mu * N $ we have $ \...
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52 views

Dirichlet series and Dirichlet convolution

Let $f$ and $g$ be an arithmetic functions, and let $f*g$ be the Dirichlet convolution of $f$ and $g$. As known from fundamental analytic number theory, the Dirichlet series generating function is: $...
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1answer
57 views

Lp Convergence of Fourier Integrals Using Hilbert Transform

I am reading about Hilbert transform and its application on Fourier analyisis, and I am triying to prove a statement given by Terence Tao in his notes on Fourier Analysis. He says that if $\varphi\in\...
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36 views

Commutative monoid structures on $\Bbb{N}$

Suppose $m \oplus n$ is a commutative and associative binary relation $\oplus: \Bbb{N} \times \Bbb{N} \to \Bbb{N}$, and that $1$ is an identity element for this operation. In other words, $(\oplus, \...
2
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1answer
81 views

A certain identity of a Dirichlet series

I have encountered this problem: I need to prove that $\sum_{n=1}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^{3}(s)}{\zeta(2s)}$. Now, I already know that $\frac{\zeta(s)}{\zeta(2s)} = \sum_{n is ...
2
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1answer
52 views

Question related to the expression of prime, twin-prime, and Sophie Germain prime counting functions in terms of Mertens function

This question assumes the following definitions. (1) $\quad\pi(x)==\sum\limits_{p\le x}1\qquad\text{(prime counting function where $p\in P$ is a prime})$ (2) $\quad\pi_2(x)==\sum\limits_{p_2\le x}1\...
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3answers
50 views

Proving identity using Dirichlet L functions

I'm trying to prove the following identity using Dirichlet L functions : ${\displaystyle \sum _{d\mid n}\varphi (d)=n}$ I have shown proved that the Dirichlet Series of $\varphi (n)$ equals to ${\...
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42 views

Convolution integral with fraction expansion

I have to solve this convolution integral $$ \int_{-\infty}^{+\infty} \frac{1}{\frac{1}{T_1} + i2\pi \tau } + \frac{1}{\frac{1}{T_2} + i2\pi (f- \tau) } d \tau $$ but I have a lot of problems with ...
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1answer
179 views

A question regarding change of index of summation in analytic number theory

I am trying exercises of Apostol's Dirichlet Series and Modular Functions in Number Theory and I am unable to get past an argument in this question. !Original Question statement of book]1 ->!...
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1answer
61 views

Proof of an identity concerning the prime $\zeta$ function

I have to prove the following identity: let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then \begin{equation} P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)). \end{equation} I proved that \...
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35 views

On solutions of $\varphi(n)=\frac{1}{2n}\sum_{1\leq d\mid n}\varphi(dn)$, where $\varphi(m)$ denotes the Euler's totient function

I wondered if one can to get easily an answer for the following question (I have thought about the other direction $\Leftarrow$). I don't know if it is in the literature, please refer it in comments ...
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1answer
121 views

Estimate for $\sum_{n\leq x}2^{\Omega(n)}$

I need some help to find a mistake in my proof. I have to prove that $\sum_{n\leq x}2^{\Omega(n)}\sim cx\log^2x$ for $x\rightarrow+\infty$, where $\Omega(p_1^{k_1}\cdot\ldots\cdot p_j^{k_j})=k_1+\...
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1answer
67 views

Question on the coefficient of the Dirichlet series related to $\frac{\zeta(s+2)}{\zeta(s)}$

This question is about the evaluation of $a(n)$ defined in (1) below which is related to the Riemann zeta function $\zeta(s)$ as illustrated in (2) below. (1) $\quad a(n)=\sum\limits_{d|n}\frac{\mu\...
2
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1answer
62 views

A symbolic formula / closed form for Dirichlet inversion?

On compting the dirichlet inverse $f^{-1}(pqrs)$ where I assumed $(p, q, r, s)$ to be arbitrary primes, I arrived at the formula: $(p, q, r, s)$ $$ f^{-1}(pqrs) = 24f(p)f(q)f(r)f(s)/f(1)^5 + \\ -6f(...
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1answer
68 views

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) $ ....
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1answer
68 views

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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0answers
43 views

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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1answer
80 views

Dirichlet Convolution of λ*u(n)

For two multiplicative arithmetic functions 𝑓,𝑔 the Dirichlet convolution is defined by $(𝑓∗𝑔)(𝑛)=∑_{𝑎·𝑏=n}=𝑓(𝑎)𝑔(𝑏). $ Let u(n) be the function that is 1 for all n and define λ(n) = $(-1)^...
3
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1answer
74 views

determining a convolution of an arithmetic function

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 $ \...
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3answers
120 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 ...
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1answer
72 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.
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1answer
40 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 ...
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2answers
114 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 : ...
3
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2answers
212 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
25 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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1answer
49 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
61 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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2answers
91 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 :)
2
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1answer
79 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 ...
3
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0answers
129 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
229 views

Inverse of completely multiplicative function with respect to dirichlet convolution

Is the inverse of a completely multiplicative function $f(n)$ with respect to Dirichlet convolution again completely multiplicative? I know that for multiplicative functions its true(Apostol's ...
2
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1answer
374 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_{...
1
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1answer
67 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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0answers
44 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
44 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
134 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
267 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 ...
2
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2answers
452 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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111 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)...