I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that

$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n \in \mathbb{N}$$

where $B_n$ is the $n$th Bernoulli number, and $\vartheta,$ the Riemann-Siegel theta function. More accurately, it seems that

$$\frac{1}{2} \left(\log \left(\frac{\left| B_{2 n}\right| }{\sqrt{2 n}}\right)-\left(\frac{2}{3}\right)^{2/3} \pi \right)$$

is closer to the actual value of $\vartheta (2n),$ and it may be true to say that

$$\lim_{n\rightarrow\infty}\frac{1}{2} \left(\log \left(\frac{\left| \sqrt{2}n \zeta(1-2n)\right| }{\sqrt{ n}}\right)-\left(\frac{2}{3}\right)^{2/3} \pi \right)=\vartheta (2n),$$

and it seems likely that large values of $|B_{2n}|$ can be estimated with

$$\sqrt{2 n} \exp \left(2 \vartheta (2 n)+\pi \left(\frac{2}{3}\right)^{2/3}\right).$$

I haven't seen any references anywhere alluding to any connection between the absolute values of the Bernoulli numbers at even $n,$ and the Riemann-Siegel theta function previously. Is the above statement correct, and if it is, what is the connection?