(T) Discovered by Eratosthenes of Cayenne 2,000 years ago in Ancient Greece, the Sieve of Eratosthenes is an algorithm that finds all prime numbers up to a specified integer:

“It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve’s key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes.”

Erica Klarreich from Quanta Magazine has written a wonderful article “a new generation of mathematicians pushes prime number barriers” about the renaissance of that topic that started in 2013 with the findings of Yitang Zhang, and a new approach from Oxford University’s Professor James Maynard, a 2022 Fields Medal recipient, and a follow-up Polymath project from UCLA’s Professor Terence Tao.

Yitang Zhang proved that you will never stop finding pairs of primes separated by at most 70 million, James Maynard that this gap can be reduced to 600, and the Polymath8 project to 246. And, assuming the generalized Elliott–Halberstam conjecture, Professor Maynard’s approach reduces the gap size to 12, while the Polymath project further reduced it to 6.

Professor Zhang published his proof in the Annals of Mathematics, “bounded gaps between primes“, Professor Maynard published his result in “small gaps between primes“. And, the result of the Polymath project was published in 2014: “new equidistribution estimates of Zhang type“.

Here is a good lecture from Professor Maynard on prime numbers:

And, on the twin prime conjecture:

But wait there is much more to come…

To estimate how many numbers a sieve removes up to some point N, mathematicians use an approach based on inclusion/exclusion that requires to know the “level of distribution” which is a number that captures how quickly prime remainders become evenly distributed into buckets, sometimes with reference to a particular type of sieves.

In 2020, Professor Maynar proposed a series of three papers which analyzed the level of distribution, and showed that it is at least 0.6, beating the previous record of 0.57 from the 1980s.

In the past few months, three of Maynard’s graduate students, Julia Stadlmann, Alexandru Pascadi, and Jared Duker Lichtman, have written papers extending both Maynard’s and Zhang’s results; one of these papers, by Jared, pushed Maynard’s level of distribution up to about 0.617. Jared then used that increase to calculate improved upper bounds on the number of twin primes up to a given stopping point, and the number of “Goldbach representations” — representations of even numbers as the sum of two primes.”

Finally, predicting how fast the buckets start to even out is also given by the generalized Riemann hypothesis. This hypothesis, if true, would imply that if we are looking at all the primes up to some very large number N, then prime remainders are evenly distributed into buckets for any divisor up to about the square root of N.

Alexandru’s findings about the level of distribution for smooth numbers and Jared’s findings about level of distribution for primes numbers reached 23% beyond the square-root barrier of the Riemann hypothesis (and 25% assuming Selberg’s eigenvalue conjecture)…

In the following video, Jared explained how his refined bound for the prime numbers and the refined bound of Alexandru for smooth numbers “Beyond the Riemann hypothesis – primes and smooth numbers“:

Note that Jared also proved the Erdős primitive set conjecture, see Quanta Magazine article about it, an Oxford University press article about it and following is a talk about it from Jared:

Here is a lecture from him on his previous work on twin primes, and a modified linear sieve:

Note: The picture above is Sungha, a wonderful dog, who may be was not a dog as he had the will and the emotions of a human being.

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