The Congruencex1x2≡x3x4(modp), the Equationx1x2≡x3x4, and Mean Values of Character Sums

✍ Scribed by Anwar Ayyad; Todd Cochrane; Zhiyong Zheng

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 748 KB
• Volume: 59
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0105

✦ Synopsis

We obtain the asymptotic formulae |B & V| =( |B|Âp)+O(-|B| log 2 p) for the number of solutions of the congruence x 1 x 2 #x 3 x 4 (mod p) in a box B of arbitrary size and position, and N(B)=(12Â? 2 ) B 2 log B+CB 2 +O(B 19Â13 log 7Â13 B), with C given explicitly, for the number of solutions of the diophantine equation x 1 x 2 =x 3 x 4 with 1 x i B. We also obtain the upper bound for fourth order character sum moments, 1Â( p&1) /{/ o | a+B x=a+1 /(x)| 4 < <B 2 log 2 p.

1996 Academic Press, Inc.

of cardinality |B|=B 1 B 2 B 3 B 4 , and V/Z 4 denote the set of integer solutions of (1). We may also view B and V as subsets of F 4 p . Solutions of (1) with some x i #0 (mod p) may be readily dealt with and so we assume henceforth that B does not meet any of the coordinate planes article no.