Vol. 15, No. 3, 1965

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ISSN: 0030-8730
A note on multiple exponential sums

L. Carlitz

Vol. 15 (1965), No. 3, 757–765
Abstract

Put

      p∑−1
S(c) =     e(x+ y +cx′y′),
x,y=1

Where e(x) = e2πi∕p and xx′≡ yy′≡ 1 (mod p), Mordell has conjectured that S(c) = O(p). The writer shows first, by an elementary argument that S(c) = O(p32). Next he proves, using a theorem of Lang and Weil that S(c) = O(p118). Finally he proves that S(c) = O(p54); the proof makes use of the estimate

p∑−1
ψ (f(x)) = O (p1∕2),
x=0

where ψ(a) is the Legendre symbol and f(x) is a polynomial of the fourth degree.

Mathematical Subject Classification
Primary: 10.41
Milestones
Received: 28 July 1964
Revised: 23 September 1964
Published: 1 September 1965
Authors
L. Carlitz