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I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$.

One of the main points of the method is to approximate the generating function in the form $$f(\rho e(\frac aq + \beta) ) \sim \frac Cq S(a, q) (1-\rho e(\beta)) ^{-1/2} \qquad (\star) $$

where $$S(a, q) = \sum_m e(am^2/q) $$

where the sum is over m modulo $q$. This lead to decomposing the sought coefficients into singular series and singular integral, up to an error term. However, I do not understand where does $(\star) $ come from. It does not seem to me to be a straightforward development of the function. Also, is it dependent on the particular choice of the problem of sum of squares, or does an analogous of $(\star) $ also hold for every function in this context? How to get a precise error term?

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