While tweaking the definition for the Euler gamma constant I found that the following appears to be true:
$$\zeta(s)=\lim_{n\to \infty } \, \frac{a^{s-1} \sum\limits_{k=1}^{a n} \frac{1}{k^s}-b^{s-1} \sum\limits_{k=1}^{b n} \frac{1}{k^s}}{a^{s-1}-b^{s-1}}$$
when $\Re(s)>0$, $a>0$ and $b>0$.
Can you prove it?
Use the AFE (approximate functional equation): $\zeta(s)=\sum_{k \le x}k^{-s}-\frac{x^{1-s}}{1-s}+O(x^{-\sigma})$ uniform in $\sigma \ge \sigma_0>0$ and valid for say $|t| < \pi x, s=\sigma+it$
Fixing $s, \Re s=\sigma >0$ and letting $n$ large enough so $x=an,bn$ satisfy the above, one has:
RHS($n$)=$\zeta(s)+O(n^{-\sigma})$ since the oscillating terms $a^{s-1}\frac{(an)^{1-s}}{1-s}-b^{s-1}\frac{(bn)^{1-s}}{1-s}$ obviously cancel out.
This implies the result by taking $n \to \infty$
Note that $a,b>0$ can be arbitrary fixed, not necessarily integral