aa r X i v : . [ m a t h . N T ] J u l An identity for the Kloosterman sum
D. I. Tolev
Abstract
We establish a simple identity and using it we find a new proof of a result ofKloosterman.Keywords: Kloosterman sums; MSC 2010: 11L05, 11L07.
The Kloosterman sum is the defined by K ( p ; a, b ) = p − X x =1 e p ( ax + bx ) , (1)where p is a prime, a and b are integers, x is the inverse of x modulo p and e p ( α ) =exp (cid:16) πiαp (cid:17) . It is clear that it takes always real values. This sum was introduced in 1926by Kloosterman [1] and he established that | K ( p ; a, b ) | ≤ / p / for p ∤ ab. (2)In 1948 A.Weil [6] improved substantially the estimate (2) and obtained the followingdeep and important inequality: | K ( p ; a, b ) | ≤ √ p for p ∤ ab. (3)Later Stepanov [5] found an elementary proof of (3) (see also Iwaniec and Kovalski [4],Chapter 11). Information about the applications of Kloosterman’s sum in analytic numbertheory as well as a simple proof of (2) can be found in Heath-Brown’s paper [3]. Anotherproof of (2) is available in the recent preprint [2] from Fleming, Garcia and Karaali.In this short note we present an identity for the Kloosterman sum and using it we finda new proof of (2) (with smaller constant in the right-hand side of this inequality).From this point onwards we assume that p > (cid:16) · p (cid:17) be theLegendre symbol. We write for simplicity K ( a, b ) = K ( p ; a, b ). Our result is the following1 heorem. For any integers a , b such that p ∤ ab we have K ( a, b ) = p + p X l =1 (cid:18) l − lp (cid:19) K ( a, lb ) . (4) Proof:
Using (1) we find K ( a, b ) = X ≤ x,y ≤ p − e p ( a ( x − y ) + b ( x − y )) = p X h =1 e p ( ah ) X ≤ x,y ≤ p − x − y ≡ h (mod p ) e p ( b ( x − y ))= p − Y ( a, b ) , (5)where Y ( a, b ) = p − X h =1 e p ( ah ) X ≤ y ≤ p − p ∤ y + h e p (cid:0) b (cid:0) y + h − y (cid:1)(cid:1) . We put y = hz in the inner sum and obtain Y ( a, b ) = p − X h =1 e p ( ah ) p − X z =1 e p (cid:0) b h (cid:0) z + 1 − z (cid:1)(cid:1) . Now we change the order of summation and use (1) to get Y ( a, b ) = p − X z =1 K (cid:0) a, b (cid:0) z + 1 − z (cid:1)(cid:1) = p − X l =1 K ( a, lb ) λ l , (6)where λ l is the number of integers z such that 1 ≤ z ≤ p − z + 1 − z ≡ l (mod p ).We easily see that λ l equals the number of solutions of the congruence lz + lz + 1 ≡ p ), hence from the properties of the Legendre symbol it follows that λ l = 1 + (cid:18) l − lp (cid:19) . (7)From (1) and our assumption p ∤ ab we get p − X l =1 K ( a, lb ) = p − X x =1 e p ( ax ) p − X l =1 e p ( blx ) = − p − X x =1 e p ( ax ) = 1 . (8)The identity (4) is a consequence of (5) – (8). (cid:3) Now we obtain immediately the following2 orollary. If p ∤ ab then | K ( a, b ) | ≤ p p + p / . (9) Proof:
Denote by Z the second term in the right-hand side of (4). From Cauchy’sinequality we get | Z | ≤ p / p X l =1 K ( a, lb ) ! / = p / Z / , say. From (1) it follows that Z = X ≤ x,y ≤ p − e p ( a ( x − y )) p X l =1 e p ( bl ( x − y )) = p ( p − . Hence | Z | ≤ p / and using (4) we obtain (9). (cid:3) Finally we mention that by the same method we can estimate also the sum K r ( p ; a, b ) = p − X x =1 e p ( ax r + bx ) , for arbitrary positive integer r . We can prove that K r ( p ; a, b ) ≪ r p / for p ∤ ab , but weshall not give the details here. Acknowledgments:
The present research was supported by Sofia University Grant172/2010.
References [1] H.D.Kloosterman,
On the representation of numbers in the form ax + by + cz + dt ,Acta Mathematica, 49, (1926), 407–464.[2] P.S.Fleming, S.R. Garcia, G.Karaali, Classical Kloosterman sums: representationtheory, magic squares, and Ramanujan multigraphs , arXiv:1004.3550.[3] D.R.Heath-Brown,
Arithmetic applications of Kloosterman sums , NAW, 5/1, 4(2000), 380–384.[4] H.Iwaniec, E.Kowalski,
Analytic number theory , Colloquium Publications, vol. 53,Amer. Math. Soc., 2004. 35] S.Stepanov
An estimation of Kloosterman sums , Izv. Akad. Nauk SSSR Ser. Mat.,35, (1971), 308-323[6] A. Weil,