TOPICS
Search

Large Sieve

The large sieve is a method in analytic number theory that bounds a sum of squares of values of a trigonometric polynomial sampled at separated points. Let M be an integer, N a positive integer, and let x_1, ..., x_R be real numbers separated by at least delta>0 modulo 1. For arbitrary complex coefficients a_n,

sum_(r=1)^R|sum_(n=M+1)^(M+N)a_ne^(2piinx_r)|^2<=(N+delta^(-1))sum_(n=M+1)^(M+N)|a_n|^2.

Here the distance between x_r and x_s modulo 1 is the distance of x_r-x_s to the nearest integer (Montgomery and Vaughan 1973).

Other forms of the large sieve bound sums of squares over families of number theoretic characters. For example, Goldmakher and Louvel (2013) proved a large-sieve inequality for quadratic characters over number fields.

See also

Analytic Number Theory, Number Theoretic Character, Sieve, Trigonometric Polynomial

Explore with Wolfram|Alpha

References

Goldmakher, L. and Louvel, B. "A Quadratic Large Sieve Inequality over Number Fields." Math. Proc. Cambridge Philos. Soc. 154, 193-212, 2013. https://doi.org/10.1017/S0305004112000370.Montgomery, H. L. and Vaughan, R. C. "The Large Sieve." Mathematika 20, 119-134, 1973. https://doi.org/10.1112/S0025579300004708.

Cite this as:

Weisstein, Eric W. "Large Sieve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LargeSieve.html

Subject classifications