- Large sieve
In
mathematics , the large sieve is a method ofanalytic number theory . As the name implies, it was developed insieve theory , (for example) sifting from an integer sequence by means of congruence conditionsmodulo prime number s in which a relatively large number ofresidue class es for each modulus are excluded. That is, a large sieve, where a proportion of residue classes is sifted out, in principle is to be distinguished from a smallsieve , in which perhaps only a single residue class for a given modulus is excluded from the sifted set. As is typical of sieve theory, this all will take place in a range of values for the parameters beyond the easy cases where theChinese remainder theorem applies to give asymptotic estimates.The early history of the large sieve traces back to work of Yu. B. Linnik, in 1941, working on the problem of the
least quadratic non-residue . SubsequentlyAlfréd Rényi worked on it, using probability methods. It was only two decades later, after quite a number of contributions by others, that the large sieve was formulated in a way that was more definitive. This happened in the early 1960s, in independent work ofKlaus Roth andEnrico Bombieri . The nature of the fundamental inequality was by then better understood: it relates toexponential sum s evaluated at points on theunit circle that are in a sense well-spaced (measured by minimum distance), and the type of inequality is derived from the principle that theoperator norm of a matrix of characters of the circle, evaluated at a finite set of points, is equal to the norm of theadjoint operator . In applications the set of points is often aFarey series of rational numbers, mapped onto the unit circle atroots of unity .ee also
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Bombieri–Vinogradov theorem References
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