I would like to estimate the absolute value of the following difference
$$ \Delta(L) = \sum_{\alpha=-L+1}^L \frac{1}{1+2 L} e^{i t \sec^2\left(\pi\frac{\alpha - 1/2}{2 L+1}\right)} - \int_{-\frac{1}{2}}^\frac{1}{2} e^{i t \sec^2(\pi x)}dx $$
where $L$ is an integer, for $L \to \infty$. I am assuming $t$ has a positive imaginary part.
Numerically, I have evidence that the error decays like $|\Delta(L)| \approx c L^{-d} e^{- a L^b}$, with $b$ approximately $\frac{1}{2}$ or $\frac{2}{3}$. I would like to find $b$ and $a$ (I'm not so interested in $d$ and $c$).
I tried to use the Euler-Maclaurin formula to estimate the error, but all derivatives of the boundary function $e^{i t \sec^2\left(\pi\frac{x - 1/2}{2 L+1}\right)}$ vanish in the strict $L \to \infty$ limit, and I'm not sure how to resum them for large $L$.