Find formulas for

\(\displaystyle\sum_{n=1}^{\infty}\frac{\sigma_s(n)}{n^s}\),

\(\displaystyle\sum_{n=1}^{\infty}\frac{\sigma_{s+1}(n)}{n^s}\)

and

\(\displaystyle\sum_{n=1}^{\infty}\frac{\sigma_{s-1}(n)}{n^s}\)

in terms of the zeta function.

Hint:Firstly, assume that \(\zeta(0)\),\(\zeta(1)\) and \(\zeta(-1)\) have a finite value and use it.Also, use that

\(D(a,s).D(b,s)=D(a•b,s)\)

and put b(n)=1.

I leave the rest to you.You can attain some pretty interesting results here.

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