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Questions tagged [dirichlet-convolution]

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

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Are there Functions $f$ and $g$ such that $g$ is Bijective & is the Dirichlet Inverse of an Arithmetic Function $f$?

Let $f: \mathbb{N} \to \mathbb{N}$ be a function. I was wondering if, there is a function $g: \mathbb{N} \to \mathbb{N}$ such that $(f * g)(n)= \epsilon (n) = (g * f)(n)$ and $(f \circ g)(n) = id(n) = ...
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Confusing Non-recursive Dirichlet Inverse Formula for Arithmetic Functions

Recall: Definition of Dirichlet Convolution and Arithmetic Functions An arithmetic function $f$ is a function $f: \mathbb{N} \rightarrow \mathbb{C}$, where $\mathbb{N} = \{1,2,3,4, \cdots\}$ and $\...
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How may I show $\sum_{d \mid n} \frac{\mu(d)^2}{\phi(d)} = \frac{n}{\phi(n)}$? [duplicate]

I wish to show the identity $$\sum_{d \mid n} \frac{\mu(d)^2}{\phi(d)} = \frac{n}{\phi(n)},$$ where $\mu$ is the Möbius function defined by $$\mu(n) = \begin{cases}(-1)^k & \text{$n=p_1 \dots p_k$,...
Robin's user avatar
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“Logarithm” with respect to Dirichlet convolution

Similar to a previous question of mine, I’m trying to find equivalents to logarithms in other rings in which we can carry out computations. If we consider an arithmetic function to be a function of ...
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Dirichlet convolution inverse of Euler's totient function

Let g(n) be the convolution inverse of Euler's totient function $\varphi(n)$. Let $n=p_1^{a_1}...p_t^{a_t}$, where $p_j$ are the distinct prime divisors of $n$. Find a formula for $g(n)$ and prove ...
lockash's user avatar
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Book reference for studying Dirichlet Convolution

Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly ...
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Discrete Fourier Transform & GCD

While reading Wolfgang Schramm's original paper concerning the relationship between the discrete Fourier transform and gcd, I came across the following condensed argument for his more general theorem ...
John Smith's user avatar
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Freshman's dream and the commutativity of the square root of the Möbius function over the divisors.

Let the infinite matrix $A$ be: $$A(n,k)=\left[ k \mid n \right] \left(\frac{\sqrt{k \, \mu(k)}}{n^s}\right)$$ where $n=1,2,3,4,5,...$ and $k=1,2,3,4,5,...$ Multiply $A$ with its transpose $A^{\mathsf{...
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Dirichlet convolutions and a formula given in Selberg's sieve

I was reading the Selberg's sieve theorem and stumbled on one equation that I honestly cannot understand. As in theorem, we let $A$ be a set of positive integers, $\mathcal{P}$ -set of primes, let $$ ...
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Convolution Method for Bound

I am reading A survey of gcd-sum functions where the following result is stated: Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by: $$\sum_{n=1}^\infty \frac{P(...
Juan Esteban Arevalo Gomez's user avatar
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?

The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$ A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p $$ is this serie calculated ...
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Characterization of Möbius-monotonicity

We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
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Which arithmetic functions satisfy $\sum_{d \mid n} f(\frac{n}{d}) k^d \equiv 0 \pmod{n}$ for all positive integers $n,k$?

There are $$\frac{1}{n} \sum_{d \mid n} \varphi\left(\frac{n}{d}\right) \cdot k^d$$ different $k$-ary necklaces of length $n$ as a result of Pólya's enumeration theorem, where $\varphi$ is Euler's ...
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Dirichlet's Series - Convergence

Calculate the expression of the following Dirichlet's series: $$ \dfrac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \dfrac{\varphi(n)}{n^s} $$ $$ \dfrac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty} \dfrac{...
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Count non-decreasing sequences of a given length N made from multiples under a limit M

Given integers N and M, count how many sequences A of N integers satisfy the following conditions? $1 ≤ A_i​ ≤ M(i=1,2,…,N)$ $A_{i+1}$ is a multiple of $A_i$. $(i=1,2,…,N−1)$ For example for N=3 and M=...
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Claim: The Euler's totient function is unique with the following property

In this post, I had an idea that $$\varphi(n) = \sum_{d \mid n} d \cdot \mu(\frac{n}{d})$$ has this property, where $\mu$ is the Mobius function. Let $\star$ be the Dirichlet convolution of functions $...
Joshua Ortiz's user avatar
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What is the Dirichlet Convolution of the identity function with itself?

If you have two identity functions, then $f(d) * g(n/d)$ would be just $dn/d = n$. Since we have an $n$ added for each divisor of $n$, would the resulting function just be $n$ times the number of ...
Isaac Wachsman's user avatar
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Term-wise product of arithmetic functions and its Dirichlet generating function

If we know Dirichlet generating function F(s) of $f(n)$ and G(s) of g(n) we can express generating function of Dirichlet convolution of $f(n)$ and $g(n)$ as product of the two generating functions $F(...
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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^...
zhrd's user avatar
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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 $n=1$ ...
Kishalay Sarkar's user avatar
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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))...
user652838's user avatar
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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 ...
jscoot's user avatar
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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: $...
José Duarte de Azevedo e Cunha's user avatar
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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 ...
tox123's user avatar
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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 ...
Phady Chergui's user avatar
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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}-...
JimSi's user avatar
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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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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 ...
Elena Martini's user avatar
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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 >...
Lorenz H Menke's user avatar
2 votes
2 answers
203 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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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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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 $ \...
Lone Learner's user avatar
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6 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: $...
Or Shahar's user avatar
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1 answer
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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\...
K. Carrillo's user avatar
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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, \...
Rivers McForge's user avatar
3 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 ...
roni 's user avatar
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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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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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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 ...
Elena Martini's user avatar
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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
108 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 \...
UnusualMathem's user avatar