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 variables, strengthening his 1924 dissertation research on five or more variables. Let a, b, m be natural numbers. Then
K ( a , b ; m ) = ∑ gcd ( x , m ) = 1 0 ≤ x ≤ m − 1 e 2 π i m ( a x + b x ∗ ) . {\displaystyle K(a,b;m)=\sum _{\stackrel {0\leq x\leq m-1}{\gcd(x,m)=1}}e^{{\frac {2\pi i}{m}}(ax+bx^{*})}.}
Here x* is the inverse of x modulo m.
Context 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.
Properties of the Kloosterman sums If a = 0 or b = 0 then the Kloosterman sum reduces to the Ramanujan sum. K(a, b; m) depends only on the residue class of a and b modulo m. Furthermore K(a, b; m) = K(b, a; m) and K(ac, b; m) = K(a, bc; m) if gcd(c, m) = 1. Let m = m1m2 with m1 and m2 coprime. Choose n1 and n2 such that n1m1 ≡ 1 mod m2 and n2m2 ≡ 1 mod m1. Then
K ( a , b ; m ) = K ( n 2 a , n 2 b ; m 1 ) K ( n 1 a , n 1 b ; m 2 ) . {\displaystyle K(a,b;m)=K\left(n_{2}a,n_{2}b;m_{1}\right)K\left(n_{1}a,n_{1}b;m_{2}\right).}
This reduces the evaluation of Kloosterman sums to the case where m = pk for a prime number p and an integer k ≥ 1. The value of K(a, b; m) is always an algebraic real number. In fact K(a, b; m) is an element of the subfield K ⊂ R {\displaystyle K\subset \mathbb {R} } which is the compositum of the fields
Q ( ζ p α + ζ p α − 1 ) {\displaystyle \mathbb {Q} \left(\zeta _{p^{\alpha }}+\zeta _{p^{\alpha }}^{-1}\right)}
where p ranges over all odd primes such that pα || m and
Q ( ζ 2 α − 1 + ζ 2 α − 1 − 1 ) {\displaystyle \mathbb {Q} \left(\zeta _{2^{\alpha -1}}+\zeta _{2^{\alpha -1}}^{-1}\right)}
for 2α || m with α > 3. The Selberg identity:
K ( a , b ; m ) = ∑ d ∣ gcd ( a , b , m ) d ⋅ K ( a b d 2 , 1 ; m d ) . {\displaystyle K(a,b;m)=\sum _{d\mid \gcd(a,b,m)}d\cdot K\left({\tfrac {ab}{d^{2}}},1;{\tfrac {m}{d}}\right).}
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