Let be the series:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \sum_{n} e^{if(n)}[/tex] where f is a function perhaps a Polynomial ..then my question is..how can this series to be evaluated (at least approximately) ?..perhaps using Euler-Bernoulli sum formula, and another question what are they used for?, i heard in a book that Goldbach conjecture could be proved using them.

Another question if we have..[tex] \sum_{n} e^{if(n)}[/tex] summed over the integers or a subset of integers..could the numbers f(n) be considered "frecuencies of vibration" or eigenvalues of a certain operator?..in fact there is an interesting connection with Physics ..if we define the partition function:

[tex] Z(u)= \sum_{n>0}e^{-uE(n)} [/tex] under "complex rotation" (u-->ix ) the partition function becomes a trigonometric sum..where in this case E(n) are the "energies" (eigenvalues) of a certain Hamiltonian.

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# Trigonometric series

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