Hirzebruch \(L\)-polynomials and multiple zeta values (Q1784161)

Let \(p_i\) be the \(i\)-th Pontryagin class of the tangent bundle of a manifold \(M\) and \(\{L_k(p_1, \ldots , p_k)\}\) be the multiplicative sequence of polynomials belonging to the power series \[ \frac{\sqrt{t}}{\tanh \sqrt{t}}=1+\frac{1}{3}t - \frac{1}{45}t^2+ \cdots +(-1)^{k-1}\frac{2^{2k}B_k}{(2k)!}t^k + \cdots. \] Here \(B_k\) denotes the \(k\)-th Bernoulli number. \(L_k\) is called the \(k\)-th Hirzebruch \(L\)-polynomial. \(L_k\) has the form \[ L_k(p_1, \ldots , p_k)=\sum h_{j_1, \ldots,j_r}p_{j_1} \cdots p_{j_r}, \] where the sum is over all partitions \((j_1, \ldots , j_r)\) of \(k\). In this paper the authors write down the coefficients \(h_{j_1, \ldots,j_r}\) of the Hirzebruch \(L\)-polynomials explicitly. As a corollary, they show that \(h_{j_1, \ldots,j_r}\) is negative if \(r\) is even and positive if \(r\) is odd. A key observation is that \(h_k\) can be expressed in terms of the alternating zeta function \(\zeta^*(s)\). They also show that similar results hold for the polynomials associated to the \(\widehat{A}\)-genus.