  We give estimates for the torsion in the Postnikov sections $\tau_{[1, n]}
S^0$ of the sphere spectrum, and show that the $p$-localization is annihilated
by $p^{n/(2p-2) + O(1)}$. This leads to explicit bounds on the exponents of the
kernel and cokernel of the Hurewicz map $\pi_*(X) \to H_*(X; \mathbb{Z})$ for a
connective spectrum $X$. Such bounds were first considered by Arlettaz,
although our estimates are tighter and we prove that they are the best possible
up to a constant factor. As applications, we sharpen existing bounds on the
orders of $k$-invariants in a connective spectrum, sharpen bounds on the
unstable Hurewicz of an infinite loop space, and prove an exponent theorem for
the equivariant stable stems.
