Questions tagged [dirichlet-convolution]

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

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A question about the property of completely multiplicative functions

I am self-studying number theory. In Apostol's number theory textbook, Theorem 2.17 states: Let $f$ be multiplicative. Then $f$ is completely multiplicative if and only if $$f^{-1}(n)=\mu(n) f(n) \...
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Product form of a Dirichlet convolution

I have derived (by a little unpure way) the product form for a Dirichlet convolution $$f*g\left(n\right)=\sum_{d|n}f\left(d\right)g\left(\frac nd \right)=\prod_{p|n} \left( \sum_{m=0}^{k_p} f \left( p^...
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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, ...
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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 ...
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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 ...
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Does equal Bell series imply Dirichlet convolution?

Apostol states in Ch.$2$ , section $2.17$ of his "Intro to Analytic NT" book; For any two arithmetical Functions $f$ and $g$ let $h = f*g$. Then for every prime $p$ we have $h_p(x)=f_p(x) \...
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$f$ is multiplicative $\implies f^{-1}$ is multiplicative. [closed]

Let $f$ be a multiplicative function i.e. $f(mn)=f(m)f(n)$ for all $m,n$ satisfying $\gcd(m,n)=1$ and $f\not\equiv 0$. Define $f^{-1}$ to be the function $g$ such that $f*g=I$ where $I(n)=1$ if $n=1$ ...
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Dirichlet inverse for $\left\{1,0,-1,0,1,0,-1,0,1,0,-1,0,\ldots\right\}$

I am looking for the Dirichlet inverse of $\left\{1,0,-1,0,1,0,-1,0,1,0,-1,0,\ldots\right\}$ or equivalently $$ f(n)=\frac{ {i^{n-1}+(-i)^{n-1}}}{2}. $$ It is an interesting inverse, it seems always ...
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Fourier series: proving that the limit is zero

Let $f: \mathbb{R}\to \mathbb{C}$ be a $2\pi$ periodic function that satisfies: $f(t)=\frac{1}{t^{\frac{1}{3}}}$ for every $t\in (0,2\pi]$. Show that: $\;\lim_{n\to \infty} \int_0^{2\pi} |f(t)-(S_n(f))...
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2 answers
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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 ...
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Is there a continuous function $f: \mathbb{T} \rightarrow \mathbb{R}$ such that $\lim_{n\rightarrow\infty} |S_{n}f(0)| = 1$?

Problem Is there a continuous function $f: \mathbb{T} \rightarrow \mathbb{R}$ such that $\lim_{n\rightarrow\infty} |S_{n}f(0)| = 1$? Relevant Definitions The Dirichlet Kernel, $D_{n}$, is defined as: $...
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2 votes
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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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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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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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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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1 vote
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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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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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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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2 votes
2 answers
107 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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1 answer
164 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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28 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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5 votes
1 answer
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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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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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1 vote
0 answers
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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, \...
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2 votes
1 answer
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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 ...
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1 answer
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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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0 votes
3 answers
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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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0 answers
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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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1 answer
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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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1 vote
1 answer
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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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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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4 votes
1 answer
184 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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2 votes
1 answer
79 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\...
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2 votes
1 answer
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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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2 votes
1 answer
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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) $ ....
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1 answer
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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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1 vote
0 answers
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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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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)^...
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3 votes
1 answer
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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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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 ...
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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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  • 561
2 votes
1 answer
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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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  • 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 : ...
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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)...
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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 ...
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1 answer
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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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-2 votes
1 answer
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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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1 vote
2 answers
108 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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