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.

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 ${\displaystyle \tau (m)}$ 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

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