The unimodality conjecture for the function beta

Let $\beta(t)$ be the function defined in the paper, and let $\mu$ and $\alpha$ be the constants used there. Unimodality conjecture for $\beta$. There exists a constant $\rho \in (\log \mu, \log \alpha)$, with $\rho \approx 1.282$, such that $\beta(t)$ is strictly increasing on $[\log \mu,\rho]$ and strictly decreasing on $[\rho,\infty)$. Moreover, $\beta(t)$ is strictly increasing on $[\log(\mu/2),\log \mu)$. This conjecture describes the expected single-peak behavior of $\beta(t)$; the preceding proposition establishes only that $\beta(t)$ has at least one local maximum in a narrower interval, so the asserted global monotonicity remains open.