Dirichlet Convolution

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Definition: let \(f,g:\mathbb{N}\rightarrow \mathbb{C}\) are two arithmetic functions, The Dirichlet Convolution \(f*g\) defined by: \[(f*g)(n)=\sum_{d|n} f(d)g\Big(\dfrac{n}{d}\Big)=\sum_{ij=n} f(i)g(j)\]

example \((i)\) Let \(\varphi (n)=|\left\{\,j\le n:\gcd(n, j)=1\right\} |\) be the Euler phi's function, \(\mu\) be the Mobius function, \(e(n) = \cases{1 & , n=1 \cr 0 & , n>1} \)

And \(\left(Tf\right)(n)=nf(n)\). Note that \(\displaystyle n=\sum_{d|n}\varphi(d)\), \(\mu * 1 = e\) where \(1(n)=1\) for any natural number \(n\).We have that,\[\varphi * 1=\sum_{d|n}\varphi(d)1\left(\dfrac{n}{d}\right)=n1(n)=T1\Longrightarrow\varphi =\varphi *e=\varphi *(1*\mu)=(\varphi * 1)*\mu=T1 *\mu\] \((ii)\) Let \(\left\{\,f_i\right\}_{i=1}^\infty \) be a sequence of the arithmetic functions, none of which is identically zero. Also assume that \(f_1*f_2*...\) converges. Prove that \(f_1*f_2*...\not = 0\).

Define \(\displaystyle \prod_{i=1}^{l*}f_i=f_1*f_2*...*f_l\) .Note that, \(\displaystyle \prod_{i=1}^{l*}f_i(n)=\sum_{n_1n_2...n_l=n}\left(\prod_{i=1}^l f_i(n_i)\right)\) As we knew \(\displaystyle\prod_{i=1}^{l*}f_i(n)\) converges as \(l\rightarrow \infty\), there exist finite indice \(i\) that \(f_i(1)=0\) , says, \(i_1,i_2,...i_\nu\in\mathbb{S}\) for the arbitrary set \(\mathbb{S}\)

None of which (arithmetic function in the sequence) is identically zero, means (In my opinion it says there is a finite number that can make the function be zero.) there exists primes \(p_{i_1},p_{i_2},...p_{i_\nu}\) correspding to the indice ,in which \(f_{i_q}(p_{i_q})\not =0\) for any \(i_q\in\mathbb{S}\)

Let \(\displaystyle n=\prod_{i\in\mathbb{S}}p_i\) We then have that, since \(f_{i_q}(1)=0\) then we have left only the following term on the right hand and \(f_i(1)\not =0\) for any number \(i\not\in\mathbb{S}\), \[\displaystyle\prod_{i=1}^{l*}f_i(n)=\left(\prod_{i\in\mathbb{S}}f_i(p_i)\right)\left(\prod_{j\not\in\mathbb{S}}f_j(1)\right)\not =0\] Hence, \(\displaystyle f_1*f_2*...=\lim_{l\rightarrow \infty}\prod_{i=1}^{l*}f_i\not=0\) as desired.

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In mathematics (in particular, functional analysis) convolution is a mathematical operation
on two functions (f and g) that produces a third function expressing
how the shape of one is modified by the other.

The term convolution refers to both the result function and to the process of computing it.
It is defined as the integral of the product of the two functions after one is reversed and shifted.

The convolution can be defined for functions on Euclidean space, and other groups.
For example, periodic functions, such as the discrete-time Fourier transform,
can be defined on a circle and convolved by periodic convolution.
(See row 18 at DTFT ยง Properties.)
A discrete convolution can be defined for functions on the set of integers.

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In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions;
it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.


The restriction of the divisors in the convolution to unitary, bi-unitary or
infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution
(existence of a Mรถbius inversion, persistence of multiplicativity, definitions of totients,
Euler-type product formulas over associated primes, etc.).

Dirichlet convolution is the convolution of the incidence algebra for the positive integers ordered by divisibility.

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