Mathematics

Questions tagged [dirichlet-convolution]

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

Dirichlet convolution is a type of convolution used in number theory for arithmetic functions. It forms a commutative ring under pointwise addition. It is defined as

\begin{equation*} (f\ast g)(n)=\sum_{d|n}f(d)g(\frac{n}{d}) \end{equation*}

where $f$ and $g$ are two arithmetic functions, and $d$ is a divisor of $n.$

Note that the multiplication of Dirichlet series is compatible with Dirichlet convolution.

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Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

I'm trying to work through Ireland and Rosen's A Classical Introduction to Modern Number Theory as I've heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius function, and $\phi$ the totient function. Find…
yunone
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Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Möbius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised Möbius inversion formula states that the above is…
Meow
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Prove that $\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$

In my notes: $$\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$$ where $\mu(d)$ is the Möbius function and $\sigma(d)$ is the sum of all positive divisors of $d$. And I have no idea how they got the expression on the right hand side. Could…
roxrook
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Prove $\sum_{k\mid n}{\mu(k)d(k)}=(-1)^{\omega{(n)}}$

I have the following exercise. Show that for all natural numbers $n$ , the following equality holds $$\sum_{d|n}{\mu{(d)}d(d)}=(-1)^{\omega{(n)}}$$ Here, $\mu$ is the Möbius function, $d$ counts the number of divisors of $n$ , and $\omega$ counts…
Lalaloopsy
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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: $\;\lim_{n\to \infty} \int_0^{2\pi} |f(t)-(S_n(f))(t)|^2 dt=0$ . We notice that $\;\lim_{n\to…
user652838
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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: $DG(f;s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ .…
Or Shahar
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When does iterated Dirichlet convolution with 1 return the original function?

Let $f : \mathbb{N}^* \to \mathbb{R}$ be a function defined on the positive integers. Define a sequence of functions $f_k$ by setting $$ f_1 = f, \quad f_{k+1}(n) = \sum_{d \mid n} f_k(d), $$ that is, $f_{k+1} = f_k * 1$ , where $*$ denotes…
Martin.s
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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(b) = \int^x_b \frac{dt}{\log t} =…
user56983
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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 formula to not include any sums over divisors. I…
Atsina
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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 $m$ (it is obvious that it is a multiplicative funtion…
user243301
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Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function. Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct prime numbers dividing n. Prove that $$\sum_{d|n,d>0}…
Pandamonium
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Closed formula for a multiple Dirichlet convolution of the 1-function with the identity

For two multiplicative arithmetic functions $f,g$ the Dirichlet convolution is defined by $(f\ast g) (n)=\sum\limits_{ab=n}f(a)g(b)$. Convoluting any arithmetic function with the $1$-function ($1(n)=1$ for all $n\in\mathbb{N}$) gives a summation…
user149890
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Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell series of $a$ to be $$…
user137301
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Why are Lambert generating functions important?

Let $(a_n)$ be a sequence of real numbers. The Lambert generating function of this sequence is defined as $$ L(x)=\sum_{n=1}^\infty a_n\frac{x^n}{1-x^n}. $$ I get why ordinary, exponential and Dirichlet generating functions are useful. But why are…
Mec
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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 the form $f: \mathbb{N}^* \to \mathbb{C}$ , the…
Gauss
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