|Prime numbers are defined by a sieve but a sieve does not define prime numbers.|
[see also prime checking without division]
Prime numbers are those natural numbers having no smaller integrally subdividing factors, ie. are not products of smaller primes, eg. 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 being the first 25 --(1 fits primal schemes but is not prime).
The sieve method of finding primes among the natural numbers begins with the smallest prime, 2, removes its infinitely many larger integer multiples from further consideration, and continues up the line in search of the next prime, which is immediately 3; and repeats this process of elimination of multiples ad infinitum ... leaving primes.
The sieve is similar in scheme to a grand product of sines of prime wave-length: sin (pi·t/p), where t is the domain scan, p is a prime; pi is 3.1415...; resulting in zeros at every n p.
A similar result is observed in the sinc function [sin x/x], a standard result in signal processing theory: the Fourier transform of a "box" shape, giving a "sombrero". The sinc function acts like a sieve for every number, and can be similarly generated by the product of a cosine and its infinitely many wave-length-doublings --thus illustrating its discovery, that it uses every frequency in its "box".
What makes this interesting to prime numbers is, that sinc uses no primes beyond 2, the doubling, while it completely covers infinitely many counting numbers: The difference in these two sieves is the sine versus cosine --simple quadrature.
A premise discovery under the title,