The Equality is that:
$\sum\limits_{n=0}^{\infty} \sum\limits_{j=1}^{k}\cfrac{1}{(kn+j)^s}=\sum\limits_{n=0}^{\infty} \cfrac{1}{(n+1)^s} $ , where s>1 .
how to show that is true?
The Equality is that:
$\sum\limits_{n=0}^{\infty} \sum\limits_{j=1}^{k}\cfrac{1}{(kn+j)^s}=\sum\limits_{n=0}^{\infty} \cfrac{1}{(n+1)^s} $ , where s>1 .
how to show that is true?
this is true
for any $k \in \mathbb{N}$ take $I_{k,n}=[kn+1 , k(n+1)]$
you can see that $\mathbb{N}^*=\cup_{n \in \mathbb{N}}I_{k,n}$
as $\frac{1}{(n+1)^s}>0 , \forall n \in \mathbb{N} $
then $\sum\limits_{n\in \mathbb{N}^*} \cfrac{1}{n^s} = \sum\limits_{n\in \mathbb{N}} \sum\limits_{p\in I_{k,n}}\cfrac{1}{p^s}$
Any number $m \in \mathbb{N}$ can be represented through quotient and remainder of division by $k$, as $m = n k + j$.