Ramanujan's Trigonometrical Sum and Relative Partitions

This paper arose out of a study of Ramanujan's trigonometrical sum C_m(n) defined by[4]

C_m(n)=Σ Σ^λn              (1)

where ρ=e^2πi/m and λ runs though a complete system of representatives of the prime residue classes mod m.