Kloosterman Sum - Properties of The Kloosterman Sums

Properties of The Kloosterman Sums

• The Kloosterman sum K(a,b; m) depends only on the residue class of a,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. The value of is always an algebraic real number. In fact K(a,b; m) is an element of the subfield which is the compositum of the fields for all odd primes p with and for with .

• If a=0 or b=0 then the Kloosterman sum reduces to the Ramanujan sum.

• Let with coprime . Choose with and . Then . This reduces the evaluation of Kloosterman sums to the case where for a prime number p and an integer .

• We have the Selberg identity

This identity was first stated by Atle Selberg and first proved by Kuznetsov by using the spectral theory of modular forms. Nowadays elementary proofs of this identity are known.

• Let m=p with p be an odd prime. Then no simple formula of K(a,b;m) is known and the Sato-Tate conjecture suggests that none exists. The lifting formulas below, however, are often as good as an explicit evaluation. If gcd(a,p)=1 one furthermore has the important transformation:

The symbol denotes the Jacobi symbol.

• Let with k>1, p prime and assume . Then K(a,b;m) = 0 unless in which case

Here for odd m is defined to be 1 if and if and is chosen in such a way that . This formula was first found by Hans Salie and there are many simple proofs in the literature.