User:Virginia-American/Sandbox/Ramanujan sum

Let

$T_q(n) = c_q(1) + c_q(2)+\dots+c_q(n)\mbox{ and }$ $U_q(n) = T_q + \tfrac12\phi(q).$

Then

$\sigma_{-s}(1)+ \sigma_{-s}(2)+ \dots+ \sigma_{-s}(n)$

$= \zeta(s+1) \left( n+ \frac{T_2(n)}{2^{s+1}}+ \frac{T_3(n)}{3^{s+1}}+ \frac{T_4(n)}{4^{s+1}} +\dots \right)$

$= \zeta(s+1) \left( n+\tfrac12+ \frac{U_2(n)}{2^{s+1}}+ \frac{U_3(n)}{3^{s+1}}+ \frac{U_4(n)}{4^{s+1}} +\dots \right)- \tfrac12\zeta(s) ,$

$d(1)+ d(2)+ \dots+ d(n)$

$= -\frac{T_2(n)\log2}{2} -\frac{T_3(n)\log3}{3} -\frac{T_4(n)\log4}{4} -\dots ,$

$d(1)\log1+ d(2)\log2+ \dots+ d(n)\log n$

$= -\frac{T_2(n)(2\gamma\log2-\log^22)}{2} -\frac{T_3(n)(2\gamma\log3-\log^23)}{3} -\frac{T_4(n)(2\gamma\log4-\log^24)}{4} -\dots ,$

$r_2(1)+ r_2(2)+ \dots+ r_2(n)$

$= \pi \left( n -\frac{T_3(n)}{3} +\frac{T_5(n)}{5} -\frac{T_7(n)}{7} +\dots \right) .$