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Exponential Sums

Exponential sums come from the Fourier transform of some pattern. The Fourier theory of addition modulo q gives q-th roots of unity, the Fourier theory of multiplication modulo q gives Dirichlet characters. The Fourier theory of addition on the real line gives ordinary Fourier series, that of multiplication gives Dirichlet series. The Fourier theory of 2 by 2 matrices can be used to investigate equations of the form ad-bc=h, and also bulk properties of the Kloosterman sums K(m,n;q) , which are Fourier transforms of the pattern of multiplicative inverses modulo q. Weyl Sums Σe(f(m)), where e(t) means exp(2Πit), and f(m) is a polynomial, are used to investigate representations of numbers in various forms. Van der Corput sums Σe(f(m)) , where f(m) is highly differentiable but not a polynomial, are used to count the number of integer points in regions with curved boundaries like xy < N; they also appear in studies of the  Riemann zeta function.

Table of solutions of xy congruent to 1 (modulo 11)

The Kloosterman sums are the finite Fourier transforms of this picture.