Large Sieve - Development

Development

Large-sieve methods have been developed enough that they are applicable to small-sieve situations as well. By now, something is seen as related to the large sieve not necessarily in terms of whether it related to the kind situation outlined above, but, rather, if it involves one of the two methods of proof traditionally used to yield a large-sieve result:

• An approximate Plancherel inequality. If a set 'S' is ill-distributed modulo p (by virtue, for example, of being excluded from the congruence classes A_p) then the Fourier coefficients of the characteristic function f_p of the set S mod p are in average large. These coefficients can be lifted to values of the Fourier transform of the characteristic function f of the set S (i.e., ). By bounding derivatives, we can see that must be large, on average, for all x near rational numbers of the form a/p. Large here means "a relatively large constant times |S|". Since, we get a contradiction with the Plancherel identity unless |S| is small. (In practice, to optimise bounds, people nowadays modify the Plancherel identity into an equality rather than bound derivatives as above.)

• The duality principle. One can prove a strong large-sieve result easily by noting the following basic fact from functional analysis: the norm of a linear operator (i.e., where A is an operator from a linear space V to a linear space W) equals the norm of its adjoint (i.e., ). This principle itself has come to acquire the name "large sieve" in some of the mathematical literature.

It is also possible to derive the large sieve from majorants in the style of Selberg (see Selberg, Collected Works, vol II, Lectures on sieves).