The composition of R.~Cohen's elements and the third periodic elements in stable homotopy groups of spheres

In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. As an application, we show that for $p\ge 7$, if $s\not \equiv 0, \pm 1 ,, mod ,p $,$n\not \equiv 1 ,, mod, 3$,$n>1$,then$\zeta_n\gamma_s$isanontrivialproductin$\pi_*(S)$byAdams-Novikovspectralsequence,where$\zeta_n$iscreatedbyR\mathrm{.}Cohen\text{cite}Co,$\gamma_s$ is a third periodic homotopy elements.