Weyl's inequality (number theory)
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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
- [math]\displaystyle{ |c-a/q|\le tq^{-2}, }[/math]
for some t greater than or equal to 1, then for any positive real number [math]\displaystyle{ \scriptstyle\varepsilon }[/math] one has
- [math]\displaystyle{ \sum_{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon}\left({t\over q}+{1\over N}+{t\over N^{k-1}}+{q\over N^k}\right)^{2^{1-k}}\right)\text{ as }N\to\infty. }[/math]
This inequality will only be useful when
- [math]\displaystyle{ q \lt N^k, }[/math]
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as [math]\displaystyle{ \scriptstyle\le\, N }[/math] 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. doi:10.1023/A:1013873301435. http://link.springer.com/10.1023/A:1013873301435.
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