I am solving polynomials with constraints on the coefficients and I get the following expression $$\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor$$.
I am looking for closed form solution if possible and particularly the asymptotic solution. I can approximate the second product of the sum as $$\lfloor N/a \rfloor$$ such that $$\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor \sim \sum_{a=1}^n \left({\lfloor N/a \rfloor}\right)^{2} = \zeta \left({2}\right) {N}^{2} + O \left({N \log \left({N}\right)}\right)$$ from https://mathoverflow.net/questions/94827/the-second-moment-of-a-sum-of-floor-functions.
Note that testing for a few cases $N = 100$ and $N = 1000$ we see $$\sum_{a=1}^N \lfloor (N + \lfloor N/a \rfloor)/a \rfloor + \lfloor (N - \lfloor N/a \rfloor)/a \rfloor \approx 2\sum_{a=1}^N \lfloor (N/a) \rfloor$$