Weyl's inequality (number theory) explained

In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies

|c - a/q|\le tq ^{-2} ,

for some t greater than or equal to 1, then for any positive real number

\scriptstyle\varepsilon one has\sum ^{M + N} _{x = M} \exp( 2 \pi if(x)) = O\left(N ^{1+\varepsilon} \left({t\over q} + { 1 \over N} + {t\over N ^{k-1} }+\right)^\right)\textN\to\infty.

This inequality will only be useful when

q < N ^{k,}

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as

\scriptstyle\le N provides a better bound.

References

• Vinogradov, Ivan Matveevich (1954). The method of trigonometrical sums in the theory of numbers. Translated, revised and annotated by K. F. Roth and Anne Davenport, New York: Interscience Publishers Inc. X, 180 p.
• Allakov. I. A.. 2002. On One Estimate by Weyl and Vinogradov. Siberian Mathematical Journal. 43. 1. 1–4. 10.1023/A:1013873301435. 117556877 . subscription.