They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926[1] when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four variables, strengthening his 1924 dissertation research on five or more variables.
Then Here x* is the inverse of x modulo m. The Kloosterman sums are a finite ring analogue of Bessel functions.
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).
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.
Up to the early 1990s, estimates for sums of this type were known mainly in the case where the number of summands was greater than √m.
Such estimates were due to H. D. Kloosterman, I. M. Vinogradov, H. Salie, L. Carlitz, S. Uchiyama and A. Weil.
The only exceptions were the special modules of the form m = pα, where p is a fixed prime and the exponent α increases to infinity (this case was studied by A.G. Postnikov by means of the method of Ivan Matveyevich Vinogradov).
In the 1990s Anatolii Alexeevitch Karatsuba developed[6][7][8] a new method of estimating short Kloosterman sums.
Karatsuba's method makes it possible to estimate Kloosterman's sums, the number of summands in which does not exceed
Various aspects of the method of Karatsuba found applications in solving the following problems of analytic number theory: Although the Kloosterman sums may not be calculated in general they may be "lifted" to algebraic number fields, which often yields more convenient formulas.
[11] The Kuznetsov or relative trace formula connects Kloosterman sums at a deep level with the spectral theory of automorphic forms.
Then one calls identities of the following type Kuznetsov trace formula: The integral transform part is some integral transform of g and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of g. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions.
[13] Weil's estimate can now be studied in W. M. Schmidt, Equations over finite fields: an elementary approach, 2nd ed.
The underlying ideas here are due to S. Stepanov and draw inspiration from Axel Thue's work in Diophantine approximation.
In fact the sums first appeared (minus the name) in a 1912 paper of Henri Poincaré on modular forms.
[4] After the discovery of important formulae connecting Kloosterman sums with non-holomorphic modular forms by Kuznetsov in 1979, which contained some 'savings on average' over the square root estimate, there were further developments by Iwaniec and Deshouillers in a seminal paper in Inventiones Mathematicae (1982).
A detailed introduction to the spectral theory needed to understand the Kuznetsov formulae is given in R. C. Baker, Kloosterman Sums and Maass Forms, vol.
Yitang Zhang used Kloosterman sums in his proof of bounded gaps between primes.