Lema (own)

[Filip Chindea]

Daca \( \gamma = \lim (H_n - \log(n)) \), sa se demonstreze elementar (la nivel OIM) estimarea

\( H_n = \log(n) + \gamma + 1/2n + \mathcal{O} \left(n^{-2}\right) \).

In consecinta, daca \( d(n) := \sharp \{ d \in \mathbb{N}^{\ast} \ : \ d | n \} \) este functia divizor, atunci sa se arate ca are loc identitatea

\( \sum_{k = 1}^n d(n) = n \log(n) + (2\gamma - 1)n + a - \sum_{k \le a} \left\{ \frac{n}{k} \right\} + \mathcal{O}(1) \),

unde \( a = \left\lfloor n \right\rfloor \).

PS.

(Cu siguranta, asta termina imediat si problema asta.)