Order of Divisor Function

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Theorem

For all $x \ge 1$:

$\displaystyle \sum_{n \mathop \le x} d \left({n}\right) = x \log x + \left({2 \gamma-1}\right) x + O \left({\sqrt x}\right)$

where $\gamma$ is the Euler-Mascheroni constant and $d \left({n}\right)$ is the divisor function.


Proof