Kloosterman sum

In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924. In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924. Let a, b, m be natural numbers. Then Here x* is the inverse of x modulo m. The Kloosterman sums are a finite ring analogue of Bessel functions. They occur (for example) in the Fourier expansion of modular forms. There are applications to mean values involving the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics. Because Kloosterman sums occur in the Fourier expansion of modular forms, estimates for Kloosterman sums yield estimates for Fourier coefficients of modular forms as well. The most famous estimate is due to André Weil and states: Here τ ( m ) {displaystyle au (m)} is the number of positive divisors of m. Because of the multiplicative properties of Kloosterman sums these estimates may be reduced to the case where m is a prime number p. A fundamental technique of Weil reduces the estimate when ab ≠ 0 to his results on local zeta-functions. Geometrically the sum is taken along a 'hyperbola' XY = ab and we consider this as defining an algebraic curve over the finite field with p elements. This curve has a ramified Artin–Schreier covering C, and Weil showed that the local zeta-function of C has a factorization; this is the Artin L-function theory for the case of global fields that are function fields, for which Weil gives a 1938 paper of J. Weissinger as reference (the next year he gave a 1935 paper of Hasse as earlier reference for the idea; given Weil's rather denigratory remark on the abilities of analytic number theorists to work out this example themselves, in his Collected Papers, these ideas were presumably 'folklore' of quite long standing). The non-polar factors are of type 1 − Kt, where K is a Kloosterman sum. The estimate then follows from Weil's basic work of 1940. This technique in fact shows much more generally that complete exponential sums 'along' algebraic varieties have good estimates, depending on the Weil conjectures in dimension > 1. It has been pushed much further by Pierre Deligne, Gérard Laumon, and Nicholas Katz.