Explicit formulas and the exceptional set in Goldbach's problem
by
Janos Pintz
Renyi Institute

We present a new approximate explicit formula for the number of Goldbach decomposition of an arbitrary even integer, containing only a bounded number of L-zeros. This formula (and other elaborated arguments) enables us to show various new results about Goldbach's Conjecture, among others the following ones.

THEOREM 1. The number of integers below X, which cannot be written as a sum of two primes, does not exceed the 2/3rd power of X, if X is large enough.

THEOREM 2. The number of integers below X, which cannot be written as a sum of at most three primes, does not exceed the 3/5th power of X, if X is large enough.

THEOREM 3. (Joint with I .Z. Ruzsa) Any large enough even integer can be written as the sum of two primes and eight powers of 2.

Date received: May 20, 2004