# Asymptotics of some infinite products

Let $(a_n)$ be a strictly increasing sequence of positive real numbers such that $\sum (1/a_n)<\infty$. Put $f(x)=\prod(1+x/a_n)$ for $x\geq 0$. By classical analysis, $f$ exhibits several good asymptotic behaviors at $+\infty$, leading to the question of whether the germ of $f$ generates a Hardy field over $\mathbb R(x)$, or even whether $f$ generates an o-minimal structure over the real field, possibly subject to making further regularity conditions on $(a_n)$. It is easy to see that if $(a_n)=(n^2)$, then $f$ is interdefinable with $e^x$ over the real field (and so o-minimality results). Via considerably more sophisticated arguments, o-minimality also results with $(a_n)=(n^s)$ for any $s\in (1,2)$. It is suspected that o-minimality should result from $(a_n)$ of sufficiently regular growth bounded by that of $(n^2)$. On the other hand, it appears that oscillation begins to occur if $a_n/n^2\to \infty$, and the more rapid the growth of $a_n/n^2$, the more detectable the oscillation. In particular, if $b>1$ and $(a_n)=(b^n)$, then $f$ does not generate a Hardy field and $(\mathbb R,+,\cdot,f)$ defines $\mathbb Z$. I will summarize the results we have obtained so far, and explain some conjectures arising therefrom. (Preliminary, and joint with Ovidiu Costin.)