[จูกัดเหลียง]

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.