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I’m continuing my series of articles for the Princeton Companion to Mathematics through the winter break with my article on distributions. These “generalised functions” can be viewed either as the limits of actual functions, as well as the dual of suitable “test” functions. Having such a space of virtual functions to work in is very convenient for several reasons, in particular it allws one to perform various algebraic manipulations while avoiding (or at least deferring) technical analytical issues, such as how to differentiate a non-differentiable function. You can also find a more recent draft of my article at the PCM web site (username Guest, password PCM).
Today I will highlight Carl Pomerance‘s informative PCM article on “Computational number theory“, which in particular focuses on topics such as primality testing and factoring, which are of major importance in modern cryptography. Interestingly, sieve methods play a critical role in making modern factoring arguments (such as the quadratic sieve and number field sieve) practical even for rather large numbers, although the use of sieves here is rather different from the use of sieves in additive prime number theory.
[Update, Jan 1: Link fixed.]