In the context of the probability theory of rare events i found myself dealing with these series of complex functions:
- $\sum_{n=1}^\infty(1+n)^{-k}z^{n^2}\\$ with z Complex and k Real.
- $\sum_{n=1}^\infty(1+n)^{-k}e^{\Lambda n\sqrt{\ln{z}}}$ with z Complex, k Real and $\Lambda$ complex parameter.
- $\sum_{n=1}^\infty(1+n)^{-k}e^{\Lambda n\sqrt{\ln{z}}+n^2\ln{z}}$ with k and $\Lambda$ as above.
With my supervisor we have found a close analogy of these series with Polylog functions, which converge for z<1. I need to put this in a more rigorous fashion because we need an integral representation for this functions to discuss the values of the parameters, but something is missing. No simple test of convergence seems to work at least to prove the absolute convergence.
