The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L\_1=\frac{1}{3}p\_1,$$
$$L\_2=\frac{1}{45}(7p\_2-p\_1^2),$$
$$L\_3=\frac{1}{945}(62p\_3-13p\_1p\_2+2p\_1^3)$$
Let $a\_k$ denote the coefficient of $p\_k$ in $L\_k$. Is there any estimate of $a\_k$? Is it true that $a\_k$ is positive and decreasing?


Checking the first 14 $a\_k$ using the formula in <http://www.mctague.org/carl/blog/2014/01/05/computing-L-polynomials/>, it is true that $a\_k$ is decreasing (and exponentially fast). I am wondering if this is true in general.



Coefficient of the top Pontryagin class in $L$-genus
The coefficient of $p\_k$ is given by


$$2^{2n}(2^{2n-1}-1)\frac{B\_n}{(2n)!} = \zeta(2n)\frac{2^{2n}-2}{\pi^{2n}},$$


see e.g. Appendix A of this older version of [Weiss](https://arxiv.org/pdf/1507.00153v3.pdf) (warning: for Weiss, the convention is that the Bernoulli numbers $B\_n$ are positive for all positive $n$). From this it is clear that it is always positive. You can presumably obtain growth estimate from growth estimates for the zeta function (or equivalently, Bernoulli numbers).


In fact, all coefficients of the $L$-polynomial can be expressed in terms of multiple zeta values, by the results of [Berglund-Bergström](https://link.springer.com/article/10.1007/s00208-018-1647-2) (and by their Corollary all $a\_k$ are positive).


