Jordan Summatory function

Now that we have seen Dirichlet's hyperbola method, we try to use it find sub-linear algorithms for different functions.

Jordan's totient function, denoted as $J_k(n)$, generalizes Euler's totient function. Let's use $\mathbf{J}_k(n)$ to denote Jordan summatory function. That is

$\mathbf{J}_k(n)=\displaystyle\sum\limits_{m=1}^nJ_k(m)$

We know that,$1*J_k(n)=n^k$, where '$*$' denotes Dirichlet convolution.

Choosing $f(n)=J_k(n)$ and $g(n)=1$, we have $h(n)=f*g=n^k$. The corresponding summatory functions are

$F(n)=\mathbf{J}_k(n)$, $G(n)=n$ and $H(n)=S_k(n)$.

Using the results obtained in Dirichlet's hyperbola method, we get

$\mathbf{J}_k(n)=S_k(n)-\displaystyle\sum\limits_{m=2}^n\mathbf{J}_k\left(\left\lfloor\frac{n}{m}\right\rfloor\right)$

$\mathbf{J}_k(n)=S_k(n)-\displaystyle\sum\limits_{m=1}^{n/(u+1)}  \left\lfloor\frac{n}{m}\right\rfloor J_k(n) - \sum_{m=2}^u \mathbf{J}_k\left(\left\lfloor\frac{n}{m}\right\rfloor\right) +u^\text{ }\mathbf{J}_k\left(\left\lfloor\frac{n}{u+1}\right\rfloor\right)$


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