Bounds for Incomplete Hyper-Kloosterman Sums

✍ Scribed by Wenzhi Luo

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 88 KB
• Volume: 75
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2340

✦ Synopsis

A new upper bound is obtained for the incomplete hyper-Kloosterman sum by means of Burgess' estimate, provided the number of variables in the sum is sufficiently large.

1999 Academic Press

In connection with the problem of expressing integers by a positive definite integral quadratic form in four variables, Kloosterman [9] introduced the exponential sum S(a, b; p)= : p&1 d=1 e \ ad+bd p + , ( 1 ) in his refinement of the Hardy Littlewood circle method, where e(x)= exp(2?ix), p 3 is a prime, ( p, ab)=1; and d stands for the multiplicative inverse of d( mod p). Kloosterman [9] also provided a non-trivial bound O( p 3Â4 ) for S(a, b; p), which was subsequently improved by Davenport [4] and Salie [12] to O( p 2Â3 ). Weil [14] established the best possible bound |S(a, b; p)| 2p 1Â2 , ( 2 ) which followed as a corollary of his proof of the Riemann hypothesis for algebraic curves over finite fields. For applications to arithmetic, one also needs non-trivial bounds for the incomplete Kloosterman sum, S(a, b; p, x)= : 1 d x e \ ad+bd p + , ( 3 )