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Here we take advantage of this essential characteristic and present evidence for the conjecture that $\\pi_k \\sim |s_k|/ \\log p_{k+1}^2$, where $\\pi_k$ is the number of primes in $s_k$; or even stricter, that $y=x^{1/2}$ is both necessary and sufficient for the prime number theorem to be valid in intervals of length $y$. 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