## 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

**, where**

*Σe(f(m))***means**

*e(t)***, and**

*exp(2Πit)***is a polynomial, are used to investigate representations of numbers in various forms. Van der Corput sums**

*f(m)***, where**

*Σe(f(m))***is highly differentiable but not a polynomial, are used to count the number of integer points in regions with curved boundaries like**

*f(m)**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.

#### References

- M. N. Huxley,
*Integer points, exponential sums and the Riemann zeta function, Number Theory for the Millennium*, Natick 2002, vol. II, 275-290. - G. R. H. Greaves, G. Harman, M. N. Huxley (ed),
*Sieves, Exponential Sums, and their Applications in Number Theory*, Cambridge 1997. - M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford 1996.