Brun’s pure sieve
In the first quarter of the twentieth century Viggo Brun developed an extension of the sieve of Eratosthenes that yielded good estimates on the number of elements of a set that are not divisible (http://planetmath.org/Divisibility) by any of the primes provided only that is “sufficiently regularly distributed” modulo these primes. That allowed him to prove that the sum of reciprocals of twin primes converges (the limit is now known as Brun’s constant), and that every sufficiently large even number is a sum of two numbers each having at most prime factors. In what follows we describe the simplest form of Brun’s sieve, known as Brun’s pure sieve.
The sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2) is based on the principle of inclusion-exclusion in the form
where is some multiplicative function of , and is small compared to . Then the estimate in (2) takes the form
If the truncated sum is a good approximation to the full sum, then this formula is an improvement over the sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2) because the sum over error terms is shorter.
Since is greater than for every and every such that , we have that
|and since the sum of a multiplicative function can be written as an Euler product, we get|
|to minimize the error term we choose , and get|
provided that .
In order to squeeze out the best upper bound on the number of primes not exceeding we have to minimize the right side of the inequality above. The nearly optimal choice is , and for a sufficiently large constant . With this choice we obtain
This inequality is stronger than the one obtained by an application of the sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2).
Example. (Upper bound on the number of twin primes.) If and , then the integers and cannot both be primes. Let be as above, and set . Then , and if . The remainder does not exceed . Like above we obtain
where denotes the number of twin primes not exceeding . Upon setting , and we obtain
This is the original result for which Viggo Brun developed the sieve now bearing his name. This result can be put in following striking form:
General combinatorial sieve
The inequality (2) was based on the observation that
where is the characteristic function of the set of integers with no more than prime factors, and is the characteristic function of the set of integers with no more than prime factors. It is possible to choose different functions and that satisfy the inequality (5) to obtain bounds similar to (4). The problem of optimal choice of these functions in general is very hard. For more detailed information on Brun’s sieve and sieves in general one should consult the monographs [1, 2, 3].
- 1 George Greaves. Sieves in number theory. Springer, 2001. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=1003.11044Zbl 1003.11044.
- 2 H. Halberstam and H.-E. Richert. Sieve methods, volume 4 of London Mathematical Society Monographs. 1974. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0298.10026Zbl 0298.10026.
- 3 C. Hooley. Application of sieve methods to the theory of numbers, volume 70 of Cambridge Tracts in Mathematics. Cambridge Univ. Press, 1976. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0327.10044Zbl 0327.10044.
- 4 Gérald Tenenbaum. Introduction to Analytic and Probabilistic Number Theory, volume 46 of Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0831.11001Zbl 0831.11001.
|Title||Brun’s pure sieve|
|Date of creation||2013-03-22 14:10:54|
|Last modified on||2013-03-22 14:10:54|
|Last modified by||bbukh (348)|