The prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$, It is denoted by $\pi{(x)}$. Using my computer I found that: Let $X$ be positive integer number $\leq 10^{9}$ then $$\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$$

Question: Is the result about hold? If $X$ be any positive integer number.

The prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$. It is denoted by $\pi{(x)}$. Using my computer I found that for any positive integer $X\leq 10^{9}$, $$\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$$

Question: Does the result hold for all positive integers $X$?