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$\pi(x)$ is the prime counting function (no. of prime within x)

For the interval $(x, x + \delta x]$, $\delta > 0$, what is the smallest integer $x_{0}$ such that for any $x >= x_{0}$, $\pi(x + \delta x) - \pi(x) > 0$ is always true?

For example, Bertrand's Postulate tells us that when $\delta = 1$, the smallest integer to make the above statement true is $x_{0} = 2$.

The following result might help: one paper by Rosser and Schoenfeld gives out two inequalities about $\pi(x)$:

$\frac{x}{\log{x}}(1 + \frac{1}{2\log{x}}) < \pi(x)$, for $x>= 59$,

and $\pi(x) < \frac{x}{\log{x}}(1+ \frac{3}{2\log{x}})$, for $x>1$

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    @Ankur, there is no requirement for $\delta$. I'm curious if we can find the smallest integer for any $\delta$.2012-09-30

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From Proposition 6.8 on pdf page 8 of DUSART, you may take $ x_0 = \max \left( 396738, \; e^{\left( \frac{1}{5 \sqrt \delta} \right)} \right). $ This is not the optimal value of your $x_0 = x_0(\delta)$ but it works.

Note: Dusart's adviser was Guy Robin, whose adviser was Jean-Louis Nicolas. It all fits.

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    @WillJagy, Thank you! This is indeed very helpful!2012-10-01
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I don't think that the answer above is correct. In fact, such an $ x_{0} $ does exist. It is a consequence of the Prime Number Theorem. One may refer to the Wikipedia article on Bertrand's Postulate: http://en.wikipedia.org/wiki/Bertrand%27s_Postulate. As for the original question, I'm not sure if bounds have been established for the size of $ x_{0} $, except in certain cases. Examples are also given in the Wikipedia article mentioned above. One of the references listed there, an article by Lowell Schoenfeld, gives the following result: For any $ n \geq 2010760 $, there exists a prime between $ n $ and $ \left( 1 + \frac{1}{16597} \right) n $. I believe that $ 2010760 $ is the smallest $ n_{0} \in \mathbb{N} $ corresponding to $ \delta = \frac{1}{16597} $. However, I haven't read the article yet, so please go ahead and read it to help me verify what I've written here.

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    @LindsayDuran: Sorry, the result in the answer about Schoenfeld is not conditional. There are conditional results in the paper but I think this result is from studying the zeros of the zeta function very carefully. It's really a three-paper series.2012-10-01