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I have found in the preprint An Exact Formula for the Prime Counting Function the following expression for a function that is $1$ whenever n is prime, and $0$ otherwise:

$$ \mathbb{1}_{n \in \mathbb{P}}=-\mu(n) \frac{\Lambda(n)}{\log n} /(\cos \pi n)^2=-\left(-\frac{2 \sin \pi n}{\pi n}\right)^2 \sum_{j=0}^{\infty} \frac{n^{2 j}}{\zeta(2 j)} \sum_{j=1}^{\infty} n^{2 j} \log \zeta(2 j) $$

$\mu(n)$: Möbius function, $\Lambda(n)$: Von Mangoldt function and $ \zeta(2 j)=-\frac{(-1)^j(2 \pi)^{2 j} B_{2 j}}{2(2 j) !}$: zeta function at the even integers ($B_{2 j}:$ Bernoulli numbers).

Question: are there any other formula for $\mathbb{1}_{n \in \mathbb{P}}$ or for the product $\mu(n) \frac{\Lambda(n)}{\log n}$?

R. S.
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    Wow, this paper claims to prove the Riemann hypothesis. Well, the math.GM section of arXiv is notoriously worse than the rest. – colt_browning Apr 12 '23 at 22:24
  • I suggest this great video by Eric Rowland about "an exact formula for the n'th prime". Though it takes hyperexponential time to actually evaluate it for any $n \ge 2$, so you'd be better off with the Sieve of Eratosthenes, and it really only shows that algebraic tools can be used as an (inefficient) programming language. There are multiple inefficient formulas you can come up with, the one shown in the video is just one example. – Daniel P Apr 12 '23 at 22:56

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Depends on what operations you are allowed to use. For example, $1_{n\in\mathbb{P}} = \mathrm{sgn}\,((n-1)!\mod n) $. See also Formulas for primes.

colt_browning
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