In analytic number theory we always consider Dirichlet series.
So I wonder if other functions can offer any insight into primes ?
Ofcourse I am not talking about turning Dirichlet series into taylor series.
Also I am not talking about a special set of primes such as twins or $1 \mod 30$ but rather all the primes. Guess that is close to the prime counting function then.
Does the function
$$f(z) = \sum_p \frac {z^p}{p^2}$$
where the sum is over all the primes $p$,
teach us anything about the primes ?
Can we reformulate the RH with (statements about) $f(z)$ ?
Today the prime number theorem is proved using the Dirichlet series $\sum \Lambda(n)/n^s$, where $\Lambda$ is the von Mangoldt function. This is the negative log derivative $-\zeta'(s)/\zeta(s)$, which allows us to use properties of $\zeta(s)$ to tell us about the behavior of $\psi(x) = \sum_{n \leq x} \Lambda(n)$.
In the early 20th century, Hardy & Littlewood were able to prove the prime number theorem using the power series $\sum_{n \geq 1} \Lambda(n)z^n$. The use of power series and trigonometric series to study questions about primes is still an active area: look up the circle method.
Once you see Dirichlet series uses to make deductions about primes, it's natural to ask if the "prime zeta-function" $\sum_p 1/p^s$ can be useful, but it really is the wrong object to regard as being of primary importance. The lesson I want to make is that merely writing down a sum over primes shouldn't be expected to make that sum helpful to study primes.
The answer by Vesslin Dimitrov to the MO question here might be inspiring. He discusses how results like the prime number theorem manifest themselves in error terms at different scales.