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I'm looking to find a nice expression for the asymptotic behaviour of $\sum_{n \le x} \frac{1}{n^p}$ (for $0 < p < 1$) as $x$ tends to infinity. Does anyone have any hints? I have a feeling that it should be some power of $\log x$ or $x^{\epsilon}$ for $\epsilon > 0$ but I'm stuck.

itsricobitches
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You may use the Euler Maclaurin expansion of the zeta function as explained here : $$\zeta(z) \sim \sum_{k=1}^N \frac 1{k^z} \color{#bb0000}{+\frac 1{(z-1)\;N^{z-1}}}\\\color{#006600}{-\frac 1{2\;N^z}+\frac z{12\;N^{z+1}}-\frac{z(z+1)(z+2)}{720\;N^{z+3}}+\frac{z(z+1)(z+2)(z+3)(z+4)}{30240\;N^{z+5}}}+\cdots$$

This gives with your notation : $$S_x(p):=\sum_{n \le x} \frac{1}{n^p}\approx \;\zeta(p)-\frac 1{(p-1)\;x^{p-1}}+\frac 1{2\;x^p}-\frac p{12\;x^{p+1}}+\frac{p(p+1)(p+2)}{720\;x^{p+3}}+\cdots$$ (this is an asymptotic expansion : if you add more terms the result will be more precise up to a certain point only, see the link)

Some numerical results (using $\zeta(p)$ and $3$ or $4$ terms of the approximation) :

\begin{array}{r|rrrr} \small S_x(p)&\small\text{approximation-3}\quad&\small\text{approximation-4}\quad&\small\text{'Exact'}\quad\quad&\small\text{Stieljes approx.}\quad\\ \hline \small S_{10}(.1)&\small 8.619333932024534&\small 8.619334186871510&\small 8.619334186106207&\small 8.619334416924156\\ \small S_{10}(.5)&\small 5.020997078843854&\small 5.020997899292666&\small 5.020997899292666&\small 5.020997908386879\\ \small S_{10}(.9)&\small 3.221142175070282&\small 3.221143042155160&\small 3.221143038249382&\small 3.221143042155502\\ \small S_{10^4}(.1)&\small 4423.009021883633985&\small 4423.009021883633985&\small 4423.009021883633985&\small 4423.009022113686631\\ \small S_{10^4}(.5)&\small 198.544645449523747&\small \small198.544645449523747&\small 198.544645449523747&\small 198.544645455556965\\ \small S_{10^4}(.9)&\small 15.688875888131209&\small 15.688875888131209&\small 15.688875888131209&\small 15.688875888131552\\ \end{array}

With not too bad results for large values of $x$ !

Let's add that $\zeta$ has a pole at $p=1$. This allow to replace $\zeta(p)$ for $p$ near of $1$ by its Stieltjes expansion : $$\zeta(p)=\frac 1{p-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n\;(p-1)^n$$ I added a column 'Stieljes approx.' obtained by replacing $\zeta(p)$ with the Stieltjes expansion limited to the finite $\sum_{n=0}^5\cdots\,$ (the absolute difference between this expansion and $\zeta$ is bounded by $4.45\;10^{-7}$).

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Raymond Manzoni
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There is a very nice article that was published in the American Mathematical Monthly about the growth rates of partial sums of infinite series. It includes the series you ask about and many, many others. You can view it for free here.

The reference is: Partial Sums of Infinite Series, and How They Grow. R. P. Boas, Jr. The American Mathematical Monthly, Vol. 84, No. 4. (Apr., 1977), pp. 237-258.

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