On a family involving R.L. Cohen’s ζ-element (II)

In 1981, Cohen constructed an infinite family of homotopy elements ζ k ∈ π * ( S ) represented by in the Adams spectral sequence, where p > 2 and k ⩾ 1. In the paper, we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζ n −1 β 2 γ s +3 is nontrivial in the stable homotopy groups of spheres π t ( s,n )− s −8 ( S ), where p ⩾ 7, n > 3, 0 ⩽ s < p − 5 and t ( s, n ) = 2( p − 1)[ p n + ( s + 3) p 2 + ( s + 4) p + ( s + 3)]+ s .