O'Reilly logo

Auxiliary Polynomials in Number Theory by David Masser

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

7

Exponential sums

The best-known of these is due to Gauss, namely

image (7.1)

with a prime p ≠ 2, for which it is fairly easy to show

image (7.2)

This resembles (6.26) in that we have a sum of p roots of unity, for which the trivial estimate p is greatly improved. As the proof of (7.2) will be imitated soon, we give it here.

In fact with the complex conjugate G we have

image

which is

where Nz is the number of solutions of x2y2 = z in By the discussion ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required