It is known, for example, https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Integral_test
$$\int_{1}^{N+1} x^{-1} dx < \sum_{i = 1}^N i^{-1} < \int_{1}^{N+1} x^{-1} dx+1$$
I am curious if this relationship generalizes to harmonic series to the power $0 < p \leq 1$.
Specifically, I am interesting in a lower-bound for
$$\sum_{i = M}^N i^{-p}$$ where $N > M$. Any help on this matter is greatly appreciated!
You can use the floor function $\lfloor x \rfloor$ to find a lower bound. Let $M,N\in \mathbb{N}$ with $M<N$ and let $p > 0$. Then for any $x\in [M, N]$, $\lfloor x \rfloor \leq x$. As a consequence, we have $$\lfloor x \rfloor^p \leq x^p,$$ or $$\frac{1}{x^p} \leq \frac{1}{\lfloor x \rfloor^p}.$$ When we integrate both sides on the interval $[M, N]$, we get $$\int_M^{N}\frac{1}{x^p}dx \leq \int_M^{N}\frac{1}{\lfloor x \rfloor^p}dx.$$ Since $\int_M^{N}\frac{1}{\lfloor x \rfloor^p}dx$ is equivalent to $\sum_{i=M}^{N} i^{-p}$, we have $$\int_M^{N}\frac{1}{x^p}dx \leq \sum_{I=M}^{N} i^{-p}.$$