An approximation for partial sums of zeta function

Question

I'd like to know if there is any approximation for the partial sum of zeta function $\sum_{i=1}^n \frac{1}{i^\beta}$ where $\beta$ is a real number less than or equal $1$,e.g., in the case $\beta = 1$, we have an approximation of $\ln (n)$.

Answer 1

In the same manner,

$$\sum_{i=1}^n\frac1{i^\beta}\sim\int_0^n\frac1{x^\beta}\ dx=\frac{n^{1-\beta}}{1-\beta}\quad;\beta<1$$

Better approximations may be done with the Euler-Maclaurin formula:

$$\sum_{i=1}^n\frac1{i^\beta}=\frac{n^{1-\beta}}{1-\beta}+\zeta(\beta)+\frac1{2n^\beta}-\frac\beta{12n^{\beta+1}}+\mathcal O\left(\frac1{n^{\beta+3}}\right)$$

if $\beta\ne1$, else,

$$\sum_{i=1}^n\frac1i=\ln(n)+\gamma+\frac1{2n}-\frac1{12n^2}+\mathcal O\left(\frac1{n^4}\right)$$

You can mess around and see how good this approximates as well as some other simpler approximations on this graph.