Divisor-square summatory bound ( $\sum_{n \leq X} d (n)^{2} ≪ X lo g^{3} (2 X)$ )

\sum_{n \leq X} d (n)^{2} ≪ X lo g^{3} (2 X)

. Partial summation consequently gives

\sum_{n \leq X} d (n)^{2} / n ≪ lo g^{4} (2 X)

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