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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.