I need a numerical approximation with low computational complexity of
$$f_n(\beta, \gamma) = \sum_{i=1}^n e^{-\beta \ln(i)+ \gamma{\ln}^2(i)}$$ for $n\approx10^6$, where $1\lt\beta\lt3$ and $0\leq\gamma<0.05$.
For $\gamma=0$, it is straighforward using the Euler–Maclaurin formula.
Would anyone have a suggestion for the general case with $\gamma > 0$?
Note that in the range of parameters I consider, the exponent is negative $-\beta \ln(i)+ \gamma{\ln}^2(i) < 0$