When I was taking a shower this problem came into my mind...

Let $f(n, s) = 1^{-s} + 2^{-s} + 3^{-s} + \cdots + n^{-s}$ be the partial sum of the $\zeta$ function.

In the cases where $s$ is a negative integer, there is the usual closed-form formula for $f(n,s)$ involving Bernoulli numbers.

However what about the cases in which $s$ is a positive integer?

For example, when $s = 1$, we have

$f(1,1) = 1$

$f(2,1) = \frac 1 1 + \frac 1 2 = \frac 3 2$

$f(3,1) = \frac 1 1 + \frac 1 2 + \frac 1 3 = \frac {11} 6$

$\cdots$

$f(10, 1) = \frac 1 1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 {10} = \frac{7381}{2520}$

Can one say anything about prime factors of the numerator and the denominator, in the final simplified fraction?

And what if $s$ is some larger integer?

Thank you very much.

p.s. A non-related beautiful paper on partial sums of $\zeta$, found when I tried to google the answer for my question: www.cecm.sfu.ca/~pborwein/MITACS/papers/borwein.ps