On $\sum_{n_1,\dots,n_k = 1}^\infty \frac{1}{(n_1+...+n_k)^p}$, $p \in \mathbb{R}^+$. Consider the "multiple harmonic series" $$\sum_{n_1,\dots,n_k = 1}^\infty \frac{1}{(n_1+...+n_k)^p}.$$ How can one study the behavior of this series for various values of $p \in \mathbb{R}^+$?

The series can be written
$$
\sum_{n=1}^\infty \frac{a_{n,k}}{n^p},
$$
where $a_{n,k}$ is the number of ways $n$ can be written as an ordered sum of $k$ positive integers. We have $a_{n,k}={n-1\choose k-1}\asymp n^{k-1}$, so the series converges if and only if $p>k$, and by expressing ${n-1\choose k-1}$ as a polynomial in $n$, the sum of the series is a rational linear combination of the Riemann zeta values $\zeta(p-k),\zeta(p-k+1),\ldots,\zeta(p)$.
