We focus on Chen’s theorem, proved in 1966 by Chen [Reference Chen5, Reference Chen6]. 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.
Here, it is interesting to note that while a lot of effort was put into making Vinogradov’s proof of Goldbach’s weak conjecture completely explicit, not much work was put into making Chen’s theorem explicit, while arguably this result is an even better approximation of Goldbach’s conjecture. The only attempt was made by Yamada [Reference Yamada14], but some mistakes can be found in the proof. (See [Reference Yamada14, (87) and (104)], where a log term appears to be missing. Also, no proof is given of the explicit version of the linear sieve that is used and this version is inconsistent with the versions in [Reference Jurkat and Richert11, Reference Nathanson12]). We also show that the following result follows readily from Theorem 1.
Theorem 2. All even numbers bigger than
$2$
can be written as the sum of a prime and the product of at most
$\exp (33)$
primes.
For the proof of Theorem 1, we draw inspiration from the work by Nathanson [Reference Nathanson12] and Yamada [Reference Yamada14]. We will now illustrate the most salient steps and results employed to obtain Theorem 1.
The proof of Chen’s theorem is based on the linear sieve, proved by Jurkat and Richert [Reference Jurkat and Richert11] and Iwaniec [Reference Iwaniec9], who were inspired by the work of Rosser [Reference Iwaniec10]. We base our work on another version obtained by Nathanson in [Reference Nathanson12] from unpublished notes by Iwaniec. Taking
$S(A,P(z))$
, a certain set of integers, sieved using
$P(z)$
, the product of certain primes, we find that for fixed
$D \in \mathbb {R}^+$
and
$s = \log D/\! \log z$
,
$$ \begin{align*}S(A, P (z)) < (F (s) + o_K (s))X + R_Q,\end{align*} $$
whenever
$s\ge 1$
, and if
$s\ge 2$
, we have a lower bound
$$ \begin{align*}S(A,P(z))> (f(s) - o_K(s))X - R_Q.\end{align*} $$
Here, X and
$R_Q$
are two specific sums and
$F(s)$
and
$f(s)$
two well-known functions, which are optimal. We can also note that it is simple to effectively bound the X term.
To obtain an explicit version of Chen’s theorem, we need an explicit version of the term
$o_K(s)$
. This was first obtained by Nathanson [Reference Nathanson12, Theorem 9.8]. He proved that we can take
$o_K (s) = (K - 1)e^{14-s}$
. To optimise the lower bound on N in Theorem 1, we need
$o_K(s)$
to be as small as possible. We obtain this by using a more computational approach, compared with Nathanson’s analytic one, and we thus reduce the term by a factor of 1000 (see [Reference Bordignon4]).
We are now left with bounding the error term
$R_Q$
that is related to the prime number theorem for primes in arithmetic progressions (PNTPAP). To bound this term efficiently, we focus on improving the version of the PNTPAP for medium-sized x, and isolate the contribution of the possible Siegel zero, given by Yamada [Reference Yamada13]. We obtain this result by proving an explicit version of the result by Goldston [Reference Goldston7], to obtain a
$\log x/ \log \log x$
saving. We also improve the Bombieri–Vinogradov style theorem for nonexceptional moduli [Reference Yamada13].
It remains to obtain a good explicit bound on the Siegel zero. To do so, we focus on improving the bound proven by Bennett et al. [Reference Bennett, Martin, O’Bryant and Rechnitzer1]. First we prove a general result that allows us to remove one of the two terms that appeared, in the previous results, in the upper bound of
$L'(\sigma , \chi )$
(see [Reference Bordignon2]). We then introduce a different technique, following from a paper of Hua [Reference Hua8] on the average of Dirichlet characters, and compute better lower bounds for
$L(1, \chi )$
to further improve the result for even characters (see [Reference Bordignon3]). As a consequence, we obtain
$$ \begin{align} \beta_0 \le 1- \frac{100}{\sqrt{q}\log^2 q}. \end{align} $$
The proof of Theorem 1 thus depends on the possible existence of an exceptional zero. We use some ideas from Yamada [Reference Yamada13] to handle this. Letting
$k_0$
be the modulus of the exceptional zero, we use two approaches that distinguish its size.
-
(1) If
$k_0$
is ‘big’, we can bound the error term
$R_Q$
using (1) as, in this case, we have
$100/\sqrt {q}\log ^2 q$
‘small’. -
(2) If
$k_0$
is ‘small’, we want to be sure that
$k_0$
does not appear in the sum in
$R_Q$
and this depends on the number of primes in
$P(z)$
. Here it is fundamental to use an inclusion–exclusion principle.
