Kloosterman sum

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In mathematics, a Kloosterman sum is a particular kind of exponential sum. For a prime number p we have

$K(a,b;p)=\sum e_p(ax+bx^*),$

where

e p (t) = e 2π i t / p

and x* is the inverse of x modulo p. The summation is taken for 1 ≤ x ≤ p − 1. 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 as opposed to five or more variables, which he had dealt with in his dissertation in 1924^[2].

1 Context
2 Estimates
3 History
4 References
5 External links

[edit] Context

These sums can also be defined for a composite number modulus, when sum is taken over all reduced residues to a given modulus m, with summand

e_m(an + bn*)

where n* denotes inverse modulo m. They (for prime modulus) are part of the theory of harmonic analysis over finite fields, being in a rough sense analogues there of Bessel functions. They turn out to have close connections with modular forms, and various analytic number theory techniques are used to provide estimates for the coefficients of modular forms starting with estimates for Kloosterman sums.

There are applications to mean values involving the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, and related topics.

[edit] Estimates

A fundamental technique of André Weil reduces the estimate

|K(a,b;p)| ≤ 2√p

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.

[edit] History

Weil's estimate can now be studied in W. M. Schmidt, Equations over finite fields: an elementary approach, 2nd. edn. (Kendrick Press, 2004). The underlying ideas here are due to S. Stepanov and draw inspiration from Axel Thue's work in Diophantine approximation.

There are many connections between Kloosterman sums and modular forms. In fact the sums first appeared (minus the name) in a 1912 paper of Henri Poincaré on modular forms. Hans Salie introduced a form of Kloosterman sum that is twisted by a Dirichlet character: such Salie sums have an elementary evaluation^[3]

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). Subsequent applications to analytic number theory were worked out by a number of authors, particularly Enrico Bombieri, Fouvry, Friedlander and Iwaniec.

The field remains somewhat inaccessible. 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. I (Kendrick press, 2003). Also relevant for students and researchers interested in the field is H. Iwaniec and E. Kowalski, Analytic Number Theory (American Mathematical Society).

[edit] References

^ Kloosterman, H. D. On the representation of numbers in the form ax² + by² + cz² + dt², Acta Mathematica 49 (1926), pp. 407-464
^ Kloosterman, H. D. Over het splitsen van geheele positieve getallen in een some van kwadraten, Thesis (1924) Universiteit Leiden
^ Hans Salie, Uber die Kloostermanschen Summen S(u,v; q), Math. Zeit. 34 (1931-32) pp. 91-109.