The aim of this paper is to improve the upper bound for the exceptional zeroes β 0 of Dirichlet L-functions. We do this by improving on explicit estimates for L ′ (σ, χ) for σ close to unity.
We focus on Chen's theorem, proved in 1966 by Chen [5,6]. We obtain the first completely explicit version of Chen's theorem and, in doing so, improve many mathematical tools that are needed for the task. We prove the following result. THEOREM 1. All even numbers bigger than exp(36) can be written as the sum of a prime and another integer that is the product of at most two primes.
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