Are there any studies about this function? $$f(x,n)=\sum_{k=1}^{n} x^{1/k}=x+x^{1/2}+x^{1/3}+x^{1/4}+\cdots +x^{1/n}$$
EDIT:
My first notes about it.
$f(1,n)=n$
$f'(1,n)=H_n$
$\int_0^1 \frac{f(x,n)}x\cdot dx=\frac{n^2+n}2$
where $f'(x,n)=\frac{d}{dx}f(x,n)$
To avoid fractional powers we can let $x=y^{n!}$ to change it into integer-powers, but the obtained polynomial does not have uniform paterm of powers.
$$f(y,n)=\sum_{k=1}^{n}y^{n!/k}=y^{n!}+y^{n!/2}+y^{n!/3}+y^{n!/4}+\cdots+y^{n!/n}$$ $$=y^{n!/n}\left(y^{n!-(n!/n)}+y^{(n!/2)-(n!/n)}+y^{(n!/3)-(n!/n)}+y^{(n!/4)-(n!/n)}+\cdots+y^{(n!/(n-1))-(n!/n)}+1\right)$$
As example let $n=4$ $$f(y,4)=y^6(y^{18}+y^6+y^2+y+1)$$
