# Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: \theta < 1 \:$ such that $\;\;\; \pi \hspace{-0.03 in}\left(x\hspace{-0.04 in}+\hspace{-0.04 in}x^{\hspace{.02 in}\theta}\right)-\pi \hspace{.03 in}(x) \: \sim \: \dfrac{x^{\hspace{.02 in}\theta}}{\log(x)} \;\;\;$ as $x$ tends to infinity".

Is there any somewhat-similar corresponding effective result for the density of primes in short intervals?

Motivation:
Such a result could yield an effective randomized reduction from subset sum to this variant of factoring.

• what is an effective randomized reduction? – Turbo May 1 '17 at 11:07

One can read off such an effective result from the work of Dudek. Indeed, examining his Section 3.3, we can get the following explicit lower bound for $x>\exp(8\times 10^{14})$ and $h=3x^{2/3}$: $$\sum_{x<p\leq x+h}\log p>h-\tfrac{1}{2}(1-10^{-3})h-\tfrac{1}{2}(1-10^{-3})h-0.0008h.$$ (Note that Dudek's definition of $E(x,h,k)$ should be multiplied by $-1$.) That is, $$\sum_{x<p\leq x+3x^{2/3}}\log p>0.0006x^{2/3},\qquad x>\exp(8\times 10^{14}).$$