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In 1988 Heath-Brown (" The number of primes in a short interval ", J. reine angew. Math. 389, 22-63) proved this theorem: Let $\varepsilon\left(x\right)\leq\frac{1}{12}$ be a non-negative function of $x$ . Then $$\pi\left(x\right)-\pi\left(x-y\right)=\frac{y}{\log x}\left(1+O\left(\varepsilon\left(x\right)^{4}\right)+O\left(\left(\frac{\log\left(\log x\right)}{\log x}\right)^{4}\right)\right)$$ uniformly for $$x^{7/12-\varepsilon\left(x\right)}\leq y\leq\frac{x}{\left(\log x\right)^{4}}.$$ I would know if exists a bound for $\varepsilon\left(x\right)$, i.e. $$\varepsilon\left(x\right)=O\left(*\right),\, x\rightarrow\infty.$$The only thing I found on the paper and on the internet is $$\varepsilon\left(x\right)\rightarrow0,\,x\rightarrow\infty$$ and this is quite obvious. Thank you.

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I found myself that the only thing we have is $\varepsilon\left(x\right)=o\left(1\right).$

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