Prime Numbers

70 Chapter 1 PRIMES!
such sums can be applied in interesting ways. Define, for odd prime p and
integers a, b, c, thesum
p−1
S(a, b, c) =
x=0
Use the Weyl relation (1.34) to prove
e 2πi(ax2 +bx+c)/p .
|S(a, b, c)| =0, p, or √ p,
and give conditions on a, b, c that determine precisely which of these three
values for |S| is attained. And here is an extension: Obtain results on |S|
when p is replaced by a composite integer N. With some care, you can handle
even the cases when a, N are not coprime. Note that we are describing here a
certain approach to the estimation of Gauss sums (see Exercises 2.27, 2.28).
Now use the same basic approach on the following “cubic-exponential”
sum (here for any prime p and any integer a):
p−1
T (a) =
x=0
e 2πiax3 /p .
It is trivial that 0 ≤|T (a)| ≤p. Describe choices of p, a such that equality
(to0orp) occurs. Then prove: Whenever a ≡ 0(modp) wealwayshavean
upper bound
|T (a)| < p 3/2 + p