Question

what are the modern process to do sieving

Answer 1

Sieving is a simple technique for separating particles of different sizes. A sieve such as used for sifting flour has very small holes. Coarse particles are separated or broken up by grinding against one-another and screen openings.Modern sieve shakers work with an electro- magnetic drive which moves a spring-mass system .

Answer 2

Sieving is performed to separate a sample based on its particle sizes by submitting it to mechanical force. The chosen sieving method determines the intensity, direction, and type of force. The sample is moved either in vertical or horizontal direction. Both movements are superimposed in tap sieve shakers.Modern sieves include the Brun sieve, the Selberg sieve, the Turán sieve, the large sieve, and the larger sieve. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools. Highlights include:

Brun's theorem, which asserts that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of the primes themselves diverges);

Chen's theorem, which shows that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (the product of two primes); a closely related theorem of Chen Jingrun asserts that every sufficiently large even number is the sum of a prime and another number which is either a prime or a semiprime. These can be considered to be near-misses to the twin prime conjecture and the Goldbach conjecture respectively.

The fundamental lemma of sieve theory, which (very roughly speaking) asserts that if one is sifting a set of N numbers, then one can accurately estimate the number of elements left in the sieve after {\displaystyle N^{\varepsilon }} iterations provided that {\displaystyle \varepsilon } is sufficiently small (fractions such as 1/10 are quite typical here). This lemma is usually too weak to sieve out primes (which generally require something like {\displaystyle N^{1/2}}iterations), but can be enough to obtain results regarding almost primes.

The Friedlander–Iwaniec theorem, which asserts that there are infinitely many primes of the form {\displaystyle a^{2}+b^{4}}.

Zhang's theorem (Zhang 2014) that there are infinitely many pairs of primes within a bounded distance. The Maynard–Tao theorem (Maynard 2015) generalizes Zhang's theorem to arbitrarily long sequences of primes.