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The exceptional set for the number of primes in short intervals. (English) Zbl 0972.11087

This paper examines properties implied by an asymptotic formula for the number of primes in a short interval \((x,x+h]\). Let \(\psi(x) = \sum_{n\leq x} \Lambda(n)\), where \(\Lambda(n)\) is the von Mangoldt function. The asymptotic formula is \[ \psi(x +h(x)) - \psi(x) \sim h(x), \quad \text{as} \;x\to \infty, \tag{*} \] which is known to be true if \(x^{7/12}\leq h(x) \leq x\) and conjectured to be true for \(x^\varepsilon \leq h(x)\leq x\) for any \(\varepsilon >0\). The authors define the exceptional set \[ E_\delta(X,h)= \{ X\leq x\leq 2X : |\psi(x +h(x)) - \psi(x) - h(x)|\geq \delta h(x)\}, \] which measures how frequently the asymptotic formula fails to hold by a proportion \(\delta\) of the main term.
The paper proves two simple properties on how the exceptional set changes as a function of \(\delta\). The authors define a function \(h(x)\) to be of type \(\theta\) if \(h(x)=x^{\theta + \varepsilon(x)}\) for some \(0<\theta <1\) where \(|\varepsilon(x)|\) is non-increasing, \(\varepsilon(x) =o(1)\), and \(\varepsilon(x+y) = \varepsilon(x) + O(|y|/x)\). Clearly functions such as \(x^\theta\) and \(x^\theta \log^\nu x\) are in this class. In the important special case that \(h(x) =x^\theta\), they denote the exceptional set as \(E_\delta(X,\theta)\) and allow \(\theta =1\). The authors prove that if \(h(x)\) is of type \(\theta\), \(X\) is sufficiently large, \(0<\delta ' <\delta \), and \(\delta - \delta ' \geq \exp(-\sqrt{\log X})\), then if \(x_0 \in E_\delta(X,h)\) we have that \(E_{\delta '}(X,h)\) contains the interval \([x_0-ch(X),x_0 + ch(X)] \cap [X,2X]\), where \(c = (\delta -\delta'){\theta\over 5}\). This is called the inertia property.
The second property, called the decrease property, is that for \(0<\theta_1 < \theta_2 <1\), \(h_i(x)\) of type \(\theta_i\) for \(i=1,2\), and \(0<\delta_1 < \delta_2\) where \(\delta_2-\delta_1 \geq \exp(-\sqrt{\log X})\), then \[ \max\big( |E_{\delta_1}(X,h_1)|, |E_{\delta_1}(\tfrac 32 X,h_1)|\big) \gg_{\theta_1} (\delta_2-\delta_1)|E_{\delta_2}(X,h_2)|, \] where \(|\;|\) of a set of real numbers denotes the Lebesgue measure of the set. These results allow one to deduce the asymptotic formula (*) from mean value theorems and conversely.
The result the authors prove is that for \(0<\theta <1\), \(h(x)\) of type \(\theta\), and \(Y=ch(X)\), then for any \(0<c<1/2\) and \(X=X(c)\) sufficiently large the mean value estimate \[ \int_{X}^{X+Y} |\psi(x +h(x))- \psi(x) - h(x)|^2 dx \leq {20\over \theta^2}Y^2 \] implies that (*) holds, and conversely.
In some applications one does not need the asymptotic formula (*), but only non-trivial estimates for the size of the exceptional set.
Letting \[ \mu_\delta(\theta) = \inf\{\xi\geq 0: |E_\delta(X,\theta)|\ll_{\delta ,\theta}X^\xi \} \] and \(\mu(\theta) = \sup_{\delta >0}\mu_\delta(\theta)\). It is well known that \(\mu(\theta) = 0\) if \({7\over 12}\leq \theta \leq 1\) and on the Riemann hypothesis this is true for \({1\over 2}< \theta \leq 1\). The authors prove using the zero-free region for the Riemann zeta-function together with density results on zeros that for sufficiently small \(\Delta >0\) and some constant \(c>0\) we have \[ \mu(\tfrac{1}{6} + \Delta) \leq 1-c\Delta, \quad \mu( \tfrac{7}{12} - \Delta) \leq \tfrac{5}{8}+ \tfrac{7}{4}\Delta +O(\Delta^2), \] and on the Riemann hypothesis \(\mu(\theta) \leq 1-\theta \) for \(0 <\theta \leq {1\over 2}\).

MSC:

11N05 Distribution of primes
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