mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Hua's lemma, named for

Hua Loo-keng Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the Peop ...

, is an estimate for

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 ...

s. It states that if ''P'' is an

integral-valued polynomial In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not ...

of degree ''k'',

\varepsilon

is a positive real number, and ''f'' a real function defined by :

f(\alpha)=\sum_^N\exp(2\pi iP(x)\alpha),

then :

\int_0^1, f(\alpha), ^\lambda d\alpha\ll_ N^

, where

(\lambda,\mu(\lambda))

lies on a polygonal line with vertices :

(2^\nu,2^\nu-\nu+\varepsilon),\quad\nu=1,\ldots,k.

References

Lemmas Analytic number theory {{mathanalysis-stub