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Dirichlet Convolution
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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#2
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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. Wiki 06 พฤศจิกายน 2019 10:26 : ข้อความนี้ถูกแก้ไขแล้ว 1 ครั้ง, ครั้งล่าสุดโดยคุณ share เหตุผล: เพิ่ม ที่มา |
#3
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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. Wiki |
หัวข้อคล้ายคลึงกัน | ||||
หัวข้อ | ผู้ตั้งหัวข้อ | ห้อง | คำตอบ | ข้อความล่าสุด |
Dirichlet convolution | analysisway | ทฤษฎีจำนวน | 1 | 15 มีนาคม 2015 10:16 |
เรื่องของ convolution | คนบ้า | คณิตศาสตร์อุดมศึกษา | 5 | 03 กันยายน 2008 17:48 |
ช่วยตั้งโจทย์ return period กับ convolution หน่อยค่ะ | caska | ปัญหาคณิตศาสตร์ทั่วไป | 0 | 18 พฤศจิกายน 2001 12:44 |
เครื่องมือของหัวข้อ | ค้นหาในหัวข้อนี้ |
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