Let $f(x): \mathbb{R} \to \mathbb{R}\_{\geq 0}$ be a smooth function. I am interested in estimating
sums of the form
$$
\sum\_{ A < n \leq B } \{ f(n)\}
$$
where $\{ c \}$ denotes the fractional part of $c \in \mathbb{R}$. Trivial upper bound is $B-A$. Are there any known non-trivial upper bounds for 'nice' $f$? say for example something like a polynomial or $f(x) = \sqrt{x}$ or $B/x$?



Average value of a fractional part of a function
You should look at Equidistribution Theory, in particular [equidistribution modulo 1](https://en.wikipedia.org/wiki/Equidistributed_sequence#Equidistribution_modulo_1). Equidistribution can be checked by confirming that $$\lim\_{n\rightarrow \infty}\sum\_{i=1}^n e^{2 \pi i f(n)}=0.$$


Then you should have $$\sum\_{A<n<B}\{f(n)\}=(B-A)/2+o(B-A)$$ which gives you an upper bound of $$|\sum\_{A<n<B}\{f(n)\}|\leq|B-A|(1/2+\epsilon)$$ for all $\epsilon>0$ and large enough $B-A$.


In particular this will hold if $f$ is a polynomial with at least one coefficient other than the constant term irrational. Also $\{x^n\}$ is equidistributed for almost all $x\in \mathbb{R\_{>1}}$


