Angel Kumchev

Department of Mathematics, University of Austin at Texas
3:30pm Wednesday June 9th, in K9509

Applications of the Large Sieve to Additive Prime Number Theory


Let E(N) denote the number of even integers 1 <= n <= N which cannot be written as the sum of two primes. Goldbach's conjecture states that E(N) = 1, but there still seems to be a long way to go until a proof is discovered. The sharpest estimate for E(N) dates back to Hardy and Littlewood (1923) and states that E(N) = O(N^(1/2 + epsilon)) for any fixed epsilon > 0. The caveat is that to get this one needs to assume the Generalized Riemann Hypothesis (GRH). The first unconditional bound was given by Montgomery and Vaughan (1975). They proved that E(N) = O(N^theta), where theta < 1 is an absolute constant (close to 1).

The above situation is quite typical in additive prime number theory: there are a number of problems in which assuming GRH we can easily obtain a non-trivial bound for the number of integers failing to have a particular kind of an additive representation. Since those problems are usually (more sophisticated) variants of Goldbach's problem, the conventional wisdom in the area is that any unconditional result should generalize the aforementioned result of Montgomery and Vaughan. It turns out that the conventional wisdom is dead wrong! The purpose of this talk is to survey the flurry of recent work (1999--2004) that has led to the realization that in most variants of Goldbach's problem one can use the large sieve to obtain unconditional results of a comparable (and often equal) strength to those that follow from GRH.