Non-trivial zeros play an important (main) role in the distribution of prime numbers.
Are there theorems in which trivial zeros play an important (main) role?



Are trivial zeros of the zeta function important?
Since you tagged this 'reference request', I'm going to answer with a theorem in my own paper *"Euler, the symmetric group, and the Riemann zeta function."* 


Let $\pi$ be a permutation in the symmetric group $S\_n$. An *ascent* is an occurrence of $\pi(j)<\pi(j+1)$ for $1\le j\le n-1$. For
example, the permutation $(24513)$ has $3$ ascents. The *Eulerian number* $\genfrac{\langle}{\rangle}{0pt}{1}{n}{k}$ is defined to be
the number of permutations in $S\_n$ with exactly $k$ ascents. (The Eulerian numbers are not to be confused with the
Euler numbers $E\_n$.) There is a surprising identity for alternating sums of Eulerian numbers: For integer $n\ge 1$ we have
$$
\zeta(-n)=\frac{ \sum\_{k=1}^n (-1)^{k} \genfrac{\langle}{\rangle}{0pt}{0}{n}{k}}{2^{n+1}(1-2^{n+1})}.
$$


Of course, $\zeta(-n)$ can be expressed in closed form in terms of the Bernoulli numbers by 
$$
\zeta(-n)=-\frac{B\_{n+1}}{n+1},
$$
so the theorem is also an identity relating Eulerian numbers to Bernoulli numbers. However, the proof is direct. It's also not not deep; it consists of identities in Concrete Mathematics by Graham, Knuth and Patashnik, along with Abel summation.


