Weyl's inequality (number theory)

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

ε

{\displaystyle \scriptstyle \varepsilon }

one has This inequality will only be useful when for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as

provides a better bound.