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# Can you figure these out?

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.

Note by Bogdan Simeonov
3 years ago