Guo and Qi recently discovered sharp bounds for the harmonic numbers (*qq.v.* [doi:10.1016/j.amc.2011.01.089](http://www.google.com/search?q=doi%3A10.1016/j.amc.2011.01.089)). For example, they show that $$H\_n < \ln(n) + \frac{1}{2n} + \gamma - \frac{1}{12n^2+\frac{6}{5}},$$ where $H\_n$ is the $n$th harmonic number and $\gamma$ is the Euler-Mascheroni constant.


**My question**: are there any similar bounds on the *generalized* harmonic numbers? By "generalized", I mean $H\_{n,r}$ where $$H\_{n,r} = \sum\_{k=1}^n \frac{1}{k^r}.$$


I am specifically interested in the case when $r = \frac{1}{2}$.



Bounds for generalized harmonic numbers
Let $f(x)=1/\sqrt{x}$. Then by Euler-Maclaurin summation,
$$ \sum\_{2\leq k\leq n} \frac{1}{\sqrt{k}} = \int\_1^n \frac{dx}{\sqrt{x}}+\sum\_{r=0}^m\frac{(-1)^{r+1}B\_{r+1}}{(r+1)!} \big(f^{(r)}(n)-f^{(r)}(1)\big)+ R$$
where $R$ is a remainder term of the form
$$R=\frac{(-1)^m}{(m+1)!}\int\_1^n B\_{m+1}(x)f^{(m+1)}(x)dx.$$
Here $B\_m$ is the $m$th Bernoulli number and $B\_m(x)$ is a periodic function of period 1 that coincides with the $m$th Bernoulli polynomial on [0,1).


Taking $m=0$, for instance, since $B\_1=-1/2$ and $B\_1(x)=${x} - $\frac{1}{2}$ we get
$$ \sum\_{2\leq k\leq n} \frac{1}{\sqrt{k}} \leq 2\sqrt{n}-2 + \frac{1}{2}\Big(\frac{1}{\sqrt{n}}-1\Big)+\frac{1}{3}\int\_1^n x^{-3/2} dx.$$
Now compute the integral and add back in the term $k=1$ to get an upper bound. 


Sharper bounds are possible choosing $m=1,2,3,\ldots$.


