Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that these formulae 1) "amount to basically defining $P_n = \sum_{x \le n} 1_{x \in P}$ for $P$ the set of primes, because they check each number up to $n$ for some property that only primes have" or are 2) incomputable (in time) since they involve many large sums and factorials. But then, the same people, when explaining the importance of the Riemann Hypothesis as a central question in pure mathematics, say that it's an important question because it has important implications for $\pi(x)$, the distribution of prime numbers. However, I believe these same two criticisms apply to the Riemann explicit formulae for primes.
In particular, my understanding is that the main relation RH has to primes is that the non-trivial Zeta zeros encode the Riemann spectrum, $\theta_i$, which gives corrective terms $C_i(\theta_i, x)$ that are used in the Riemann explicit formulae $\pi(x) = R_0(x) + \sum_{i=1}^\infty C_i(\theta_i, x)$. But we can see that this very formula that motivates the importance of RH has the same epistemic problems of other formulae for $P_n$, namely that it is 1) incomputable (in time -- it is an infinite sum), and 2) essentially takes the data of a function we know encodes primality (the Zeta function) and plugs that data (the Riemann spectrum/Zeta zeros) into an infinite sum to compute the number of primes $\pi(x)$. This is exactly what we criticized as rendering existing formulae (such as Willan's) as "worthless!" So how is Riemann's explicit formulae any better than those other formulae we already have, and if it isn't, why is the Riemann Hypothesis touted as having profound implications for our understanding of the prime numbers (in particular compared to existing formulae)?
