Order of Divisor Counting Function

Theorem
For all $x \ge 1$:


 * $\ds \sum_{n \mathop \le x} \map {\sigma_0} n = x \log x + \paren {2 \gamma - 1} x + \map \OO {\sqrt x}$

where:
 * $\gamma$ is the Euler-Mascheroni constant
 * $\map {\sigma_0} n$ is the divisor counting function.