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.

#InfiniteSum #RiemannZetaFunction #Goldbach'sConjurersGroup #TorqueGroup #DirichletSeries

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$