Arithmetic functions of particular type

Question

Any there any natural functions real valued single variable that:

changes (increases) values only at primes but otherwise stay constant (like a non periodic increasing staircase)?

whose increase in value at primes depend on the value of the primes (and hence the increases are also irregular like the spacing of the staircase).

the magnitude of jumps reduces/increases with increasing primes.

So I am not looking for summation of delta functions at primes.

Would there be any utility of such a natural function to number theory?

Question: How does irregularity of primes contribute to errors using any counting function? What characteristic does one need in the counting function to reduce estimate errors?

Answer 1

There is also the prime counting function:

$$\pi(x)=|\{p\,\mathrm{prime} : p\le x\}|.$$

Answer 2

There are exact formulas for various weighted prime counting functions. One of them is Chebyshev's prime counting function $$ \psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$

The sum is taken over the non-trivial zeros $\rho$ of the zeta function. This is a staircase-like function, with jumps at the primes and prime powers.

Answer 3

Chebyshev's $\vartheta$ function might work.

$$\vartheta(x)= \sum_{p\leq x} \log p. $$

It does not jump at prime powers and does not require calculation of $\rho.$

The counting function that Chebyshev worked with is

$\psi(x) = \sum \Lambda (x)$ in which $\Lambda(x)$ counts powers of primes.

This is different from von Mangoldt's formula $\psi_0(x)$ although they are closely related. Von Mangoldt's formula is $\psi(x)-\Lambda(x)$ at prime powers and $\psi(x)$ otherwise.