Asymptotics for $f(x)$ such that $f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$?

Question

Consider

$$f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$$

$$f(n) < \pi(n+1)$$

Where $\pi$ is the prime counting function and that inequality follows for instance from these here

$f(x) + f(x/2) + f(x/3) + ... = x$ and conjecture A : $\pi(x/a) > \frac{\pi(x)}{a}$

$f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$ and $\lim_{n \to \infty} \frac{f(n)}{\pi(n)} = 1$?

Now the question here is simple. What is a good asymptotic for $f(x)$ ? In particular for large (positive) $x$ ?

Maybe something like

$$\frac{x}{\ln(x) + A \sqrt{\ln(x)}+B}$$

Or

$$\frac{x}{\ln(x) + C \ln\ln(x)-1} $$

For some constants $A,B$ or $C$ ?

(The second estimate was given elsewhere(outside of MSE and MO) as a guess).

Answer 1

(the previous answer was wrong and I deleted it)

This is copy from the answer here : $f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$ and $\lim_{n \to \infty} \frac{f(n)}{\pi(n)} = 1$?

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Let $x$ be a positive integer.

I considered thinking about estimating

$$\pi(x) + \pi(x/2) + \pi(x/3) + ... $$

The idea is simple.

we take primes $p_i < x$.

and we take $2 p_i < x$

and in general

$$n p_i < x $$

Then we naively expect all numbers between $1$ and $ x $ to be of these forms.

Hence the unions of $ n p_i$ should give about $x-1$.

However consider say $6 = 2*3$ and $35 = 5*7$.

$6$ is both of the form $ 2 p_i $ and $ 3 p_i $. Similar for $35$ ; $ 5 p_i$ and $ 7 p_i$.

So it appears

$$\pi(x) + \pi(x/2) + \pi(x/3) + ... = \sum_1^x fact(i) = T(x)$$

where $fact(x)$ is the number of factorizations of $x$

Since

$$f(x) + f(x/2) + f(x/3) + ... = x$$

This implies (even without conjecture $A$ but obviously with conjecture $A$)

$$ \pi(R(x) - 1) < f(x) < \pi(x+1) $$

Where $R(x)$ is the functional inverse of $T(x)$

Based on the average number of factorizations for a random $x$, $R(x)$ has as asymptotic $\frac{x}{\ln(\ln(x))}$,

therefore

$$f(x) = \frac{\pi(x)}{\ln(\ln(x))}$$

is a good asymptotic.

This implies the limit from the OP is not true. What was probably the reason the idea got abandonned.

As for getting better estimates I see two routes at the moment:

1. Improve the estimate of $R$ or $T$.

2. Use the estimate for $f(x)$ together with the ratio's $\pi(x/m)/f(x/m)$ and the functional equation, and repeat that idea for a sequence of increasing improved asymptotics.