prime number theorem and prime counting function $\pi(x)$ 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 $x>1$

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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