The Twin Prime Conjecture, which states that there are infinitely many pairs of primes with difference 2, is one of the more difficult open questions in the theory of numbers. Sieve methods, which have their origins in the work of the ancient Greek mathematicians Eratosthenes, have been developed and refined in an attempt to resolve the Twin Prime Conjecture and related problems. Recently, Goldston, Pintz, and Yildirim have introduced some interesting new ideas in sieve methods, and they have used them to make an important advance on the Twin Prime Conjecture. These same authors, together with the investigator, have also proved results about short gaps between numbers between semi-primes; i.e., between numbers with exactly two prime factors. The sieve techniques developed by Goldston et alia have potential applications. Motivated by these techniques, the investigator proposes work on three projects. The first project is to improve sifting limits and sifting bounds for sieves of dimension greater than 2. The second project is to resolve questions on an idealized problem involving sifting by primes in limited ranges. The third project is to extend recent results on equal values of the number of divisors function for consecutive integers.
This project is for work in the area of number theory. Number theory is the study of the special properties of the integers. In particular, one of the central problems of number theory is to understand the interplay between the additive structure and the multiplicative and additive structure of the integers. Prime numbers form the basic building blocks of the integers under multiplication, but there are many open questions about the behavior of the primes with respect to addition. For example, Goldbach's Conjecture, which was first stated in 1742, is that every even number exceeding 2 is a sum of two primes. De Polignac's Conjecture, which dates to 1849, is that every even number is a difference of two primes in infinitely many ways. Both of these conjectures are open, but attempts to solve them have lead to the development of sophisticated sieve methods. In turn, sieve methods have lead to important advances in our understanding of the integers. Sieves also have practical applications within computer science; for example, the sieve of Eratosthenes is a standard benchmark test for computer software. In 1994, Thomas Nicely was studying Brun's Constant, which is the sum of the reciprocals of twin primes. His attempts to approximate this constant ultimately led him to discover a flaw in the Pentium FPU.
