Lema (own)
Posted: Thu Jul 10, 2008 4:07 pm
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.)
\( H_n = \log(n) + \gamma + 1/2n + \mathcal{O} \left(n^{-2}\right) \).
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.)