  It was proved by Chern, Hirzebruch and Serre that the signature of a fibre
bundle is multiplicative if the fundamental group of the base acts trivially on
the cohomology ring of the fibre with real coefficients, in which case the
signature of the total space equals the product of the signatures of base and
fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the
signature is multiplicative modulo 4. In this paper we present two results
concerning the multiplicativity modulo 8: firstly we identify the obstruction
to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Pontryagin
squaring operation. Furthermore, we prove that if the fibre is even-dimensional
and the action of the fundamental group of the base is trivial on the middle
cohomology of the fibre with $\mathbb{Z}_4$ coefficients, then this
Arf-Kervaire invariant takes value 0 and hence the signature is multiplicative
modulo 8.
