Consider a set of transcendental equations for $x \in (0,1):$
$$ f_i^s(x)=i^{\frac{s}{\log_i(x_i)}}=x_i, $$
for $i=1,2,3,\cdot \cdot \cdot$
Summing the solutions yields:
$$ \sum\limits_{i=1}^\infty x_i=\zeta(\sqrt{s})= \sum_{n=1}^\infty \frac{1}{n^{\sqrt{s}}}.$$
What's wrong with this argument? Is summing solutions together useful outside of differential equations, where the general solutions can be found by adding solutions together via the superposition principal?
Is there a set of equations that when solved and solutions added together yield $\zeta(s)?$
Edit:
$$f_i^{s^2}(x)=i^{\frac{s^2}{\log_i(x_i)}}=x_i.$$
$$\sum\limits_{i=1}^\infty x_i=\zeta(s).$$