While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p\_{b}-p\_{a}}{b-a}$ where $a<b$ are positive integers:


$$H\_{m}-\log\_{2} m-2\gamma$$


where $m:=\frac{a+b}{2}$, $\gamma$ is Euler-Mascheroni's constant, $\log\_{2}x=\log\log x$ and $H\_{x}=\{x\}H\_{\lceil x\rceil}+(1-\{x\})H\_{\lfloor x\rfloor}$ is a rather natural generalization of harmonic numbers.


Writing $\dfrac{p\_{b}-p\_{a}}{b-a}=H\_{m}-\log\_{2} m-2\gamma+R\_{m}$, is it true that $R\_{m}=O\_{\varepsilon}(\dfrac{1}{m^{1/2+\varepsilon}})$?


Thanks in advance.



Has this formula about prime gaps already been conjectured and/or proven?
Even the weaker statement that $\frac{p\_b-p\_a}{b-a}$ tends to infinity with $a,b\to\infty$ is false, because there are infinitely many bounded prime gaps (Yitang Zhang's theorem).


