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# On power free numbers

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Entry 1. Let $f(r)$ be any divergent series of positive terms, $q_{r,k}$ be the $r^{th}$ k-power free number, $\zeta(k)$ be the Riemann Zeta function. Let $S_f(r) = \sum_{i=1}^{r} f(i)$. If  $g(x)$ is Riemann integrable in $(0, \infty)$ then,

$\displaystyle{ \sum_{r=1}^{n} f(q_{r,k}) g \Big(\frac{S_f(r)}{\zeta(k)}\Big) \sim \int_{f(1)}^{\frac{S_f(n)}{\zeta(k)}} g(x)dx.}$

Corollary 1. As $k \rightarrow \infty, \zeta(k) \rightarrow 1$. Also every natural number is k-power free when $k \rightarrow \infty$. Hence the above result reduces to

$\displaystyle{ \sum_{r=1}^{n} f(r) g(S_f(r))\sim \int_{f(1)}^{S_f(n)} g(x)dx}$.

Entry 2. Further let $q'_{k,n}$ be the $n^{th}$ k-power containing number and $f$ be any function Riemann integrable in $(1,\infty)$; then,

$\displaystyle{ \sum_{k=2}^{\infty}\frac{1}{k} \Big\{\frac{f(q'_{k,1}) + f(q'_{k,2}) + f(q'_{k,3}) +\ldots}{f(q_{k,1}) + f(a_{k,2}) + f(q_{k,3}) +\ldots}\Big\}= 1 - \gamma}$.