Asymptotic behaviour of infinite product of ceilings $\lceil\alpha\rceil \lceil2\alpha\rceil \lceil3\alpha\rceil\dots$, $\alpha\in(0,1)$ I am interested in computing the value of a complex expression (see here) containing the following building block
$$
\prod_{j=1}^{n}\frac{j}{\lceil{\alpha j}\rceil},
$$
and its partial product starting from $j=i$, where $\alpha\in(0,1)$ is a parameter. I am interested in the behavior as $n\rightarrow\infty$. While for all $\alpha\in(0,1)$, the above expression is unbounded as $n$ grows, I would want to understand the asymptotic behavior of this infinite product parametrized by $\alpha$.

*

*Has anyone ever encountered similar expressions?

*Any ideas of how to get the asymptotic?

*Also pointers to connections with other functions in combinatorics/number theory would be useful.

Note: the fundamental difficulty arises from understanding the behavior of the term on the denominator:
$$
\lceil\alpha\rceil \lceil2\alpha\rceil \lceil3\alpha\rceil ...\lceil n\alpha\rceil,
$$
as the numerator can be simply written as $(n+1)!$
Thank you :-)

Hint:
$$\alpha j\le\lceil \alpha j\rceil<\alpha j+1$$
and
$$\alpha^nn!\le\prod_{j=1}^n\lceil\alpha j\rceil<\prod_{j=1}^n(\alpha j+1).$$
Then
$$\prod_{j=1}^n\left(\alpha+\dfrac1j\right)<\prod_{j=1}^n\left(1+\dfrac1j\right)=n+1$$
so
$$\alpha^nn!\le\prod_{j=1}^n\lceil\alpha j\rceil<(n+1)!$$
