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Primes in short intervals. (English) Zbl 0569.10023
For which unbounded, monotonic functions $\Phi(x)\leq x$ is it true that $$\pi(x+\Phi(x))-\pi(x)\sim \Phi(x)/\log x\qquad(x\to \infty)\, ? \tag{*}$$ On the Riemann hypothesis one may take any $\Phi(x)\geq x\sp{1/2+\varepsilon}$ (for a fixed $\varepsilon >0)$. One would conjecture indeed that the range $\Phi(x)\geq x\sp{\varepsilon}$ is admissible. Selberg showed, on the Riemann hypothesis, that if $\Phi(x)(\log x)\sp{-2}\to \infty$ then (*) holds for “almost all” $x$. That is to say, there is an exceptional set ${\cal E}\sb{\Phi}$ such that the Lebesgue measure of ${\cal E}\sb{\Phi}\cap [0,x]$ is $o(x)$ as $x\to \infty$, and such that (*) holds for $x\not\in {\cal E}\sb{\Phi}$. The present paper shows in an ingenious way that ${\cal E}\sb{\Phi}$ can be non-empty, and indeed that (*) does not hold for any function $\Phi(x)=(\log x)\sp{\lambda}$ with constant exponent $\lambda$. Specifically it is shown that $$ \pi(x+\Phi(x))-\pi(x)=\Phi(x)/\log x+\Omega\sb{\pm}(\Phi (x)/\log x) $$ for $\Phi(x)=(\log x)\sp{\lambda}$. The proof uses ideas from the author’s work on chains of gaps between consecutive primes [Adv. Math. 39, 257--269 (1981; Zbl 0457.10023)].

11N05Distribution of primes
11N13Primes in progressions
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