number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, Weyl's inequality, named for

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

, states that if ''M'', ''N'', ''a'' and ''q'' are integers, with ''a'' and ''q''

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

, ''q'' > 0, and ''f'' is a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

of degree ''k'' whose leading coefficient ''c'' satisfies :

, c-a/q, \le tq^,

for some ''t'' greater than or equal to 1, then for any positive real number

\scriptstyle\varepsilon

one has :

\sum_^\exp(2\pi if(x))=O\left(N^\left(+++\right)^\right)\textN\to\infty.

This inequality will only be useful when :

q < N^k,

for otherwise estimating the modulus of the

exponential sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typic ...

by means of the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...

\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. * {{Cite journal, last=Allakov, first=I. A., date=2002, title=On One Estimate by Weyl and Vinogradov, url=http://link.springer.com/10.1023/A:1013873301435, journal=Siberian Mathematical Journal, volume=43, issue=1, pages=1–4, doi=10.1023/A:1013873301435, s2cid=117556877 Inequalities Number theory