About the RH seen as a minimization problem, see the Nyman-Beurling-Baez-Duarte criterion. The major idea with this approach is that we don't even need to mention the primes.
We start from
RH is true iff for every $\Re(s) > 1/2$ : $$\lim_{N \to \infty} 1-\zeta(s)\sum_{n=1}^N \mu(n) n^{-s} = 0$$
that we can generalize with
RH is true iff there is a sequence of Dirichlet polynomials $A_N(s) =\sum_{n=1}^N a_{n,N} n^{-s}$ such that for every $\Re(s) > 1/2$ : $$\lim_{N \to \infty} 1-\zeta(s)A_N(s) = 0$$ the convergence being locally uniform,
which leads to
$$\lim_{N \to \infty} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{|1- \zeta(s)A_N(s)|^2}{|s(s-1)|} ds = 0, \qquad (\sigma > 1/2)$$
And with some work, looking carefuly at $\sum_{n=1}^N \mu(n) n^{-1/2-it}$ under the assumption that RH is true, we obtain that $\sigma = 1/2$ works, ie.
RH is true iff there is a sequence of Dirichlet polynomials such that $$\lim_{N \to \infty} \int_{-\infty}^{\infty} \frac{|1- \zeta(1/2+it)A_N(1/2+it)|^2}{t^2+1/4} dt = 0$$