Question: Do there exist any Riemann zeta $\zeta(s)$ like functions $f_N(s)$ that may have all nontrivial zeros (verified via numerical calculation) on the critical line but only involve integers up to $N$?
The Riemann zeta function $\zeta(s)$ involves all the integers (for $s>1$) and thus all the primes. I would think that if only integers up to $N$ are involved in these functions $f_N(s)$, then only finite number of primes up to $N$ can be involved.
We would also require that $$\lim_{N\to\infty} f_N(s)=\zeta(s)$$
Thus we may study the interaction of prime $2$ with prime $3$ if we set $N=3$ in $f_N(s)$. We may understand better how and at what $N$ the chaotic behavior (if it exists) of the normalized spacing between adjacent zeros of $f_N(s)$ kicks in.
Of course the Euler product may not exist anymore for these functions.
thanks- mike