given the nice entry in http://mathworld.wolfram.com/TotientSummatoryFunction.html isn't it cute to guess that Sum[Sum[EulerPhi[k],{k,1,n}],{n,1,w}] = Sum[EulerPhi[k]*(w+1-k),{k,1,w}] = (2*w^3+3*w^2+4*w)/(2*Pi^2) numerical test: Table[w=2^ww; {Sum[EulerPhi[k]*(w+1-k),{k,1,w}], Floor[(w*(4+w*(3+2*w)))/(2*Pi^2)]},{ww,16}] gives: { {3,1}, {13,9}, {75,63}, {497,457}, {3649,3482}, {27833,27196}, {217521,215001}, {1719835,1709899}, {13679291,13639043}, {109111251,108952364}, {871617967,870980213}, {6967839977,6965289383}, {55722310869,55712110761}, {445696933143,445656078836}, {3565248463365,3565085421581}, {28520683400617,28520030576058} } not too bad, no? W.