  We study the distribution of prime numbers under the unlikely assumption that
Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} \Lambda(n)
\Lambda(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$.
Such an asymptotic formula has been previously obtained only for fixed $h$ in
which case our result quantitatively improves those of Heath-Brown (1983) and
Tao and Ter\"av\"ainen (2021).
  Since our main theorems work also for large $h$ we can derive new results
concerning connections between Siegel zeros and the Goldbach conjecture and
between Siegel zeros and primes in almost all very short intervals.
