For $x, \lambda > 0$, define
$$S\_\lambda(x) := \#\{p\_{n+1} \leq x : p\_{n+1} - p\_n \geq \lambda \log x\} ,$$
where $p\_n$ is the $n$th prime number. It is known [1] that an uniform version of the Hardy-Littlewood prime k-tuples conjecture implies that for fixed $\lambda > 0$,
$$S\_\lambda(x) \sim e^{-\lambda} \frac{x}{\log x}$$
as $x \to +\infty$.


My question is: If we want only a lower bound of the form
$$S\_\lambda(x) \gg\_\lambda \frac{x}{\log x}, \quad x > 2,$$
for every fixed $\lambda > 0$, has this been proved unconditionally?


Thank you for any reference or suggestion.


[1] Funkhouser,Goldston, Ledoan, Distribution of Large Gaps Between
Primes, <https://doi.org/10.1007/978-3-319-92777-0_3>



Positive proportion of logarithmic gaps between consecutive primes
The inequality $S\_\lambda (x)\gg\_{\lambda} \frac{x}{\log x}$ should follow from the case $m=0$ of the paper <https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/positive-proportion-of-short-intervals-containing-a-prescribed-number-of-primes/BE20070F6B2864B2EFD65C268267BC5A>


at least for small values of $\lambda$. More precisely, the result should be that there exists a large constant $C>0$ such that for any $\lambda<\frac{1}{C^4\log^2C}$, we have the desired lower bound for $S\_{\lambda}(x)$. 


Although the above cited paper could be more useful to analyze the case of general gaps between primes, for gaps of consecutive primes I would like to refer you to the following result <https://www.ams.org/journals/tran/2010-362-05/S0002-9947-09-05009-0/>


in which the requested inequality may be shown for values of $\lambda<\lambda\_0$, for a certain $\lambda\_0>1$ explicitly computed in that paper. 


I doubt it has already been proven such lower bound unconditionally for any $\lambda>0$. 


