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This paper gives an explicit version of Selberg's 1943 mean-value estimatefor the prime number theorem in intervals under the Riemann hypothesis. Twoapplications are given: for primes in short intervals, and Goldbach numbers(sums of two primes) in short intervals. Under the Riemann hypothesis, we showthere exists a prime in $(y,y+32277\log^2 y]$ for at least half the$y\in[x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696\log^2 x]$ for all $x\geq 2$.

An explicit mean-value estimate for the PNT in intervals