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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 $-\beta \ln(i)+ \gamma{\ln}^2(i) < 0$

vitamin d
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Loic Thibaut
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    Welcome to Cross Validated and thanks for your question. Is there a statistical or random aspect to your question? For example could you treat $\beta$ as a truncated exponential random variable? (truncated to the indicated range and assume $\gamma$ is fixed) – Lucas Roberts Jun 23 '21 at 02:12
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    Thanks for the comment. The only statistical aspect here is that I use this function as an empirical model of the $p_i$ in a multinomial likelihood that I numericaly maximize to find ML estimates of $\beta$ and $\gamma$ – Loic Thibaut Jun 23 '21 at 02:18
  • ok, then it sounds that $f_n$ defines a likelihood function to be maximized with respect to $\beta$ and $\gamma$. Have you tried putting this through a maximization routine in your programming language of choice? I'm unclear as to why you need an approximation. – Lucas Roberts Jun 24 '21 at 00:28
  • It is purely a computational constraint: the model has a large number of parameters (20,000 disctinct values of $\beta$) and the optimization routine would take too long to converge. Furthermore summing a large number of small values is numericaly unstable. The approximation solves both problem for $\gamma = 0$, so I hoped I could do the same in the general case – Loic Thibaut Jun 24 '21 at 06:14

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