This is admittedly, probably an easy question for the right person here, but I cannot seem to track down an answer. The question itself may not be hard, but the answer is crucial to a math paper I am writing (and I don't know enough number theory).



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> True or false: There are at least exp$(r)$ irreducible polynomials in $\mathbb{F}\_2[x]$ of degree $r$ or less. [I don't care about the hidden constants in the exp$(r)$-notation; could be $2^{\frac{r}{100}}$ or it could be $\frac{2^r}{r}$.]
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If True a reference would be terrific. 


I am aware that there are roughly $n/\log n$ primes of size $n$ or less for $n$ large, I am trying to adapt that to a polynomial ring where the coefficients are in a characteristic 2 field.



Number of irreducible polynomials of degree $r$ in $F_2[x]$
Sloane's OEIS sequence [A001037](https://oeis.org/A001037) counts ($n=r$ in your definition):


Number of degree-$n$ irreducible polynomials over $GF(2)$; 


number of $n$-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period $n$; 


number of binary Lyndon words of length $n$.


The first few terms of the sequence are (for $n=1,2,...$ ) $2,1,2,3,6,9,...$


The formula for the sequence is $$\frac{1}{n}\sum\_{d|n}\mu(n/d)\cdot 2^d$$. Since the terms of the sum corresponding to strict divisors $d$ of $n$ are much smaller this almost gives what you want (leading term is $2^n/n=O(2^{n-\log n})$.


