I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that $$\mathcal{I}=~~~ \frac{1}{1-n}; n\leq0$$ \begin{align*} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{1-n}-\zeta(1/n)~~;n\\in(0,1) \end{align*} \begin{align*} ~~~~~~~~~~~1-\gamma~; n=1 \end{align*}

Where $\gamma$ is the Euler-Mascheroni constant. And $\zeta(s)$ is the Riemann Zeta function.

My questions are

1) Is anything similar known? Any other definite integrals relating the fractional part and the Riemann zeta function?

2) It is apparent from the above that $\zeta(1/n) < \frac{1}{1-n}$ for $n\\in(0,1)$. Now I've found out that the same inequality holds even when $n>1$. However, the same technique for evaluation for $\mathcal{I}$ doesnt work when $n>1$, as the computation depends on the sum $$\sum_{n=1}^\infty \frac{1}{n^p}$$ which diverges when $p\leq 1$. And if $n>1$, we have that $1/n<1$ so one of the sums in the process of evaluation becomes divergent.

I'm guessing that some complex analysis is required to overcome this difficulty. But I'm not familiar with that as of now.

I'd be grateful for any comments on this.

Thank you. :)