## 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 $$\Phi(x)\geq x^{1/2+\varepsilon}$$ (for a fixed $$\varepsilon >0)$$. One would conjecture indeed that the range $$\Phi(x)\geq x^{\varepsilon}$$ is admissible. Selberg showed, on the Riemann hypothesis, that if $$\Phi(x)(\log x)^{-2}\to \infty$$ then (*) holds for “almost all” $$x$$. That is to say, there is an exceptional set $${\mathcal E}_{\Phi}$$ such that the Lebesgue measure of $${\mathcal E}_{\Phi}\cap [0,x]$$ is $$o(x)$$ as $$x\to \infty$$, and such that (*) holds for $$x\not\in {\mathcal E}_{\Phi}$$.
The present paper shows in an ingenious way that $${\mathcal E}_{\Phi}$$ can be non-empty, and indeed that (*) does not hold for any function $$\Phi(x)=(\log x)^{\lambda}$$ with constant exponent $$\lambda$$. Specifically it is shown that $\pi(x+\Phi(x))-\pi(x)=\Phi(x)/\log x+\Omega_{\pm}(\Phi (x)/\log x)$ for $$\Phi(x)=(\log x)^{\lambda}$$. 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)].

### MSC:

 11N05 Distribution of primes 11N13 Primes in congruence classes 11N35 Sieves

Zbl 0457.10023
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