# The asymptotic form of Hermite's constant in lattice

The are some linearly upper bounds on Hermite's constant $$\gamma_d$$, such as $$\gamma_d \leq 2d/3$$, $$\gamma_d \leq d/4+1$$. So we can claim that $$\gamma_d=O(d)$$. There is also a rather tight asymptotical bound for $$\gamma_n$$: $$\frac{d}{2\pi e}+\frac{\log(\pi d)}{2\pi e}+o(1) \leq \gamma_d \leq \frac{1.744d}{2 \pi e}(1+o(1))$$ (see page 34 of The LLL algorithm edited by Phong Q. Nguyen et al.). After this tight asymptotical bound, there is a sentence that "Thus, $$\gamma_d$$ is enssentially linear in $$d$$" in this book (also in page 34). My questions is that can we express $$\gamma_d$$ in the asymptotical form of $$\Theta(d)$$. I have searched some related papers, but I can't find the the corresponding expression.

• What you want seems to follow from the inequality you have linked? The first inequality implies that $\gamma_d = \Omega(d)$. The second inequality implies that $\gamma_d = O(d)$. Combined you have that $\gamma_d =\Theta(d)$.
– Mark
Aug 2, 2023 at 5:39