Hilbert-Kunz/Limit/Brieskorn Singularitäten/en/Bemerkung

We consider the Brieskorn singularities

${}S=\mathbb {Z} /(p)[x_{1},\ldots ,x_{s}]/{\left(x_{1}^{d_{1}}+x_{2}^{d_{2}}+\cdots +x_{s}^{d_{s}}\right)}\,$

together with the maximal ideal. Also in this case it is known (Monsky-Han) that the Hilbert-Kunz multiplicities are rational and that they converge to a rational number for ${}p\mapsto \infty$. There exists an explicit but not easy description for this number. If all exponents are ${}2$, then the limit is

$1+{\text{coefficient of }}z^{s-1}{\text{ in the power series expansion of }}\operatorname {sec} \,z+\operatorname {tan} \,z.$