Primes in Almost All Short Intervals
نویسندگان
چکیده
It is well known that Huxley’s density estimates [5] for the zeros of the Riemann zeta-function yield J(x, h) = o(xh2(log x)−2), but only for h ≥ x1/6(log x) , for some C > 0. The weaker result with h ≥ x1/6+ε is proved in Saffari and Vaughan [8], Lemma 5, and in [13], where an identity of Heath-Brown (Lemma 1 of [3]) is used. This paper is inspired by Heath-Brown’s extension [4] of Huxley’s Theorem [5] that π(x)− π(x− h) ∼ h(log x)−1 to the range h ≥ x7/12−ε(x). This was achieved by means of another identity (see (2.2) of [4], or Lemma 2 below), thereby avoiding a direct appeal to the
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