Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article

Question

In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2.

On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg Martin, Prime Number Races, The American Mathematical Monthly, vol. 113, (2006), that is labeled as formula $(3)$: the first formula of the section Riemann’s revolutionary formula.

Question. Is it possible to interpret the equivalence de Reyna-Toulisse state similar to the formula or conjecture that Granville and Martin show as formula $(3)$? Many thanks.

That I am asking is if it is possible/feasible to write a similar formula that is showed by Granville and Martin,which is equivalent to the Riemann hypothesis in the same spirit as de Reyna and Toulisse show?

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In the Spanish version [2], the formula is the formula $(3)$ in page 212.

References:

[1] Juan Arias de Reyna and Jérémy Toulisse, The $n$-th prime asymptotically, Journal de Théorie des Nombres de Bordeaux, Volume: 25, Issue: 3 (2013).

[2] Andrew Granville and Greg Martin, Carreras de números primos, La Gaceta de la RSME, Volumen 8, Número 1 (enero-abril, 2005).

Answer 1

Theorem 6.2 from [1] is probably more closely analogous to a different fact stated in [2], namely the assertion (page 9) that RH is equivalent to $$ \big| \log\big( \mathop{\rm lcm}[1,2,\dots,x] \big) - x \big| \le 2\sqrt x(\log x)^2, $$ which in turn is known to be equivalent to $$ | \pi(x) - \mathop{\rm li}(x)| \ll \sqrt x\log x. $$ Theorem 6.2 is, to oversimply, the result of inverting the function $\mathop{\rm li}(x)$ (using the obvious fact $\pi(p_n)=n$).

Formula (3) from [2] is the "explicit formula" for the prime-counting function. To get an analogue of such a formula for $p_n$, it would be essentially necessary to invert the function on the right-hand side of (3)—which seems extremely challenging to say the least.