##### How many prime numbers of $b$ bits are there?


Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser and Schoenfeld [RS62, Corollary 3] prove:



> 
> *For $λ≥20.5$, $\frac{3}{5}λ/\lnλ < π(2λ) - π(λ) < \frac{7}{5}λ/\ln(λ)$ where $π(x)$
> denotes the number of primes $≤x$.*
> 
> 
> 


From this, one immediately derives that



> 
> *For $b≥5$, the number of $b$-bit primes is between $\frac{3}{5\ln 2} 2^b/b \simeq 0.865⋅2^b/b$ and $\frac{7}{5\ln 2} 2^b/b\simeq 2.02⋅2^b/b$.*
> 
> 
> 


Experimentally for small values of $b$, one gets these values for $α = \frac{π(2^{b+1})-π(2^b)}{2^b/b}$:


$$\begin{array}{l|llllllllllllllllllllllllllllll}
b& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\
α & 1.00 & 1.00 & 0.750 & 1.25 & 1.09 & 1.22 & 1.26 & 1.34 & 1.32 & 1.34 & 1.37 & 1.36 & 1.38 & 1.38 & 1.39 & 1.39 & 1.39 & 1.40 & 1.40 & 1.40 & 1.41 & 1.41 & 1.41 & 1.41 & 1.41 & 1.41 & 1.41 & 1.41 & 1.42 & 1.42
\end{array}$$


My question is about explicit (and proven!) bounds that would be closer to the experimental data:


* It *seems* that for $b≥4$, the value of $α$ is always $≥1$: Is it true and proved?
* ~~The value of $α$ *seems* to increase with $b$ from $b = 5$: Is it true and proved?~~ False, *cf.* $b = 9$ or $b=12$ for instance.
* Rosser and Schoenfeld prove that $α ≤ 2.02$: Are there improvements? Is $α ≤ 1.5$ true?


[RS62]: J. Barkley Rosser and Lowel Schoenfeld. Approximate formulas for some function of prime numbers. Illinois J. Math. 6(1): 64-94 (March 1962). DOI: [10.1215/ijm/1255631807](http://doi.org/10.1215/ijm/1255631807).



Explicit bounds on number of primes of given size
Dusart proved in his [thesis](https://www.unilim.fr/laco/theses/1998/T1998_01.pdf) that
$$\frac{x}{\log x-1}<\pi(x)<\frac{x}{\log x-1.1},\qquad x\geq 60184.$$
It follows after a bit of calculation that
$$0.975\frac{x}{\log x}<\pi(2x)-\pi(x)<\frac{x}{\log x},\qquad x\geq 2^{31}.$$
Hence for $b\geq 31$, the value of $\alpha$ lies between $0.975/\log 2\approx 1.4066$ and $1/\log 2\approx 1.4427$.


