I'm curios about the following question, from an informational viewpoint. What is the purpose in finding/getting analytic formulas for specific arithmetic functions in the context of the Riemann Hypothesis?
In the literature concerning the Riemann Hypothesis are analytic formulas for different arithmetic functions.
Example 1. The so-called explicit formula is an example of the formula for the second Chebyshev function, involving a summation over all non-trivial zeros of the Riemann Zeta function. See it as (9) in this MathWorld.
Example 2. Also, there was in the literature the reasoning for the Mertens function, see this MathWorld due to Titchmarsh, and also Odlyzko and te Riele.
Imagine that I've got a friend that tells me that he/she wants to find an analytic formula (involving a summation over the nontrivial zeros of $\zeta(s)$) for a specific arithmetic function related to some equivalence to the Riemann Hypothesis. What should be my analysis of such a situation? See my question.
Question Is there some general purpose in the attempt to find analytic formulas for specific arithmetic functions that appear in equivalences to the Riemann Hypothesis? Is the reasoning that such analytic representations should solve the Riemann Hypothesis, or will provide us valuable information about the Riemann Hypothesis? For what arithmetic functions should be interesting to find such analytic representation? I am asking/looking for an answer from an informative viewpoint, thus calculations aren't required or that your answer explicitly shows such formulas. What I am waiting for is an explanation of the why it is interesting to find such analytic representations in the context of the Riemann Hypothesis, and your explanation from an informative viewpoint. Many thanks.