Siegel Zeros, Twin Primes, Goldbach’s Conjecture, and Primes in Short Intervals

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