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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.