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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 \(\Phi(x)\geq x^{1/2+\varepsilon}\) (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

Citations:

Zbl 0457.10023
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