# Questions tagged [dirichlet-convolution]

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

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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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### 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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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 $h_p(x)=f_p(x) \... • 1,863 -2 votes 1 answer 84 views ###$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$... • 3,619 0 votes 1 answer 89 views ### Dirichlet inverse for$\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 ... • 558 6 votes 1 answer 461 views ### 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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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 ...