Taking the partial sum of the $\zeta $ function:
and the partial product f the $\zeta $ function:
I have two questions:
(1) For $p$ being prime and $n,k,i\in\Bbb N$; do there exist closed functions for $\zeta^H(s,k)$ and $\zeta^P(s,j)$ to which the partial sum respectively the partial product converge?
(2) Does anybody know from a proof that would confirm that for a given $s$, if and if only $j,k\to\infty$ then: $$\zeta^H(s,k)-\zeta^P(s,j)\to0$$
Thanks for your help in advance.