Order of Divisor Function

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Theorem

For all $x \ge 1$:

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

where $\gamma$ is the Euler-Mascheroni constant and $\map d n$ is the divisor function.


Proof