The Goldbach conjecture asks if we can always write an even number greater than or equal to 4, as a sum of two prime numbers. This will imply that all integers at least 6 can be written as a sum of three primes. The latter for odd integers ≥ 7 is a weaker conjecture and has recently been proven by Harald Helfgott. The even number case remains unsolved.
We introduce the conjecture in a quantitative form due to G.H. Hardy and J.E. Littlewood in 1919 (with details published in 1923). A weaker version of this conjecture implies the non-existence of ``exceptional zeros” of certain L-functions. This is joint work with John B. Friedlander, Daniel A. Goldston and Henryk Iwaniec.
We also introduce the notion of non-vanishing regions of the Riemann zeta function and its connection to the prime number theorem. The analogue for Dirichlet L-functions also holds except for a possible real zero which we call an ``exceptional zero”.
Counting the number of Goldbach representations itself is difficult, and taking its average tells us a bit more information. In fact, the asymptotic formula for the average number of Goldbach representations is very closely related to the quantitative form of the Prime Number Theorem (PNT) and the error is determined by a non-vanishing region of the Riemann zeta-function. This is obtained in the student project with Keith Billington and Maddie Cheng from San Jose State University, together with Jordan Schettler.
