On Ramanujan's arithmetical function \(\varSigma_{r,s}(n)\). (Q1439335)

Es sei \(n > 0\) und ganz, \(r\) und \(s\) positiv, ganz und ungerade, \[ \begin{gathered}\sigma _s(n)=\underset{d\!/n\;}{\varSigma } d^s,\\ \sigma _s(0)=\tfrac{1}{2}\,\zeta (-s),\\ \varSigma _{r,s}(n)=\sigma _r(0)\,\sigma _s\,(n)+ \sigma _r(1)\,\sigma _s(n-1)+\dots +\sigma _r(n)\,\sigma _s(0), \end{gathered} \] \[ \multlinegap{0pt} \begin{multlined} R_{r,s}(n)=\varSigma _{r,s}(n) \frac{\varGamma (r+1)\varGamma (s+1)}{\varGamma (r+s+2)}\cdot \frac{\zeta (r+1)\,(\zeta (s+1))}{\zeta (r+s=2)}\cdot \sigma _{r+s+1}(n)\\ -\frac{\zeta (1-r)+\zeta (1-s)}{r+s} \cdot n\cdot \sigma _{r+s-1}(n).\end{multlined} \] \textit{Ramanujan} (Collected papers 18) vermutete \[ R_{r,s}(n)=O(n^{\frac{1}{2}(r+s+1+\varepsilon )}) \] für jedes positive \(\varepsilon \) und \[ \varSigma _{r,s}(n)\sim\frac{\varGamma (r+1)\varGamma (s+1)}{\varGamma (r+s+2)}\cdot \frac{\zeta (r+1)\,\zeta (s+1)}{\zeta (r+s+2)}. \sigma _{r+s+1}(n). \] Verf. beweist mit der \textit{Hardy-Littlewood}schen Methode (\textit{Farey} dissection) \[ R_{r,s}(n)=O(n^{\frac{1}{2}(r+s)+1}). \]