Order of Divisor Count 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 count function.

Proof

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