Span of lens spaces

We prove that the span (i.e., maximal number of linearly independent vector fields) of $(2n + 1)$-dimensional lens spaces ${L^n}(p)$, where $p$ is an odd prime and where $n + 1 = m \cdot {2^t}$ ($m$ odd), is equal to the span of $(2n + 1)$-dimensional sphere if $t + 1 \equiv 0,1,2\pmod 4$ or if $n > 3,t + 1 \equiv 3\pmod 4$, and $p \geqq t + 3$. This result is an improvement of a theorem given by T. Yoshida.