On the divisor sum function (Q1073081)

The authors prove a Ramanujan type identy for the function \(\sigma_ r(n,k)=\sum_{d_ 1...d_ k=n}d^ r_ 1\) for \(k=3,4\). They have shown that if the real parts of s, s-a, s-b, s-a-b are all \(>1\), then for \(k=3\) \[ \sum^{\infty}_{n=1}\sigma_ a(n,k) \sigma_ b(n,k) n^{- s}=\zeta^ 3(s) \zeta^ 2(s-a) \zeta^ 2(s-b) \zeta (s-a-b) \prod_{p}F(p^{-s}) \] where p is prime and \(F(x)=1+x-(2p^ b+2p^ a+4p^{a+b})x^ 2+2(p^ a+p^ b+2)p^{a+b} x^ 3-p^{2a+2b}(x^ 4+x^ 5).\) A similar formula is proved for \(k=4\). Using the identity, they prove that \[ \sum_{n\leq x}\sigma_ a(n,3) \sigma_ b(n,3)\sim C x^{a+b+1} \] where C is a constant. Also, using the related result of A. Selberg, the authors obtain some asymptotic formulas for the summatory function of \(\sigma_ r(n,k)\).