Here is a equation obtained from Discrete Fourier transform,
$$F(k) = \frac{1}{n+1}\sum_{l=0}^{n}P^{-lk} \prod_{j=1}^{n}(1+(P^{l-1})p_j)$$ where $p_j=\frac{x_j}{N}, $$ k\in(1,2,3.....n)$, $ x_j\in(1,2,3.....N)$, $P=\exp^{\frac{2i\pi}{n+1}}$, $i=\sqrt{-1}$ and N is a constant.
Trying to estimate an asymptotic bound involving $n$ and $N$ using Euler-Maclaurin Summation. Is it correct approach to use Euler-Maclaurin Summation for asymptotic upper bound? Second, if I can estimate upper bound on $P=\exp^{\frac{2i\pi}{n+1}}$, would that be sufficient to calculate a bound on $F(k)$.
Any other methods, solutions and suggestions are welcome ! Seems very hard to find upper bound on this equation.