On Linnik theorem on Goldbach numbers in short intervals and related problems

Linnik proved, assuming the Riemann Hypothesis, that for any $\varepsilon> 0$, the interval $[N, N+\log^{3+\varepsilon}N]$ contains a number which is the sum of tw primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C\log^2 N$, the added new ingredient being Selberg's estimate for the mean-square of primes in short intevals. Here we give another proof of this sharper result which avoids the use of Selberg's estimate and is therefore more in the spirit of Linnik's original approach. We also improve an unconditional result of Lavrik's on truncated forms of Parseval's identity for exponential sums over primes.