Let $K=\mathbb Q(\zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ that can be represented by the norm form $N$ (in other words, there is an integral element of $K$, $f$, such that the field norm of $f$ is $l$) (assuming x is finite).
(I) More specifically to this problem, let $n$ be a prime and $K=\mathbb Q(\zeta_n)$ as above. Let $L(x)$ represent the number of integers less than $x$ which can be represented by the norm form $N$ also defined above. What is a good upper and lower bound for $L(2^n)$ as $n$ goes to infinity?
My conjecture (II) $2^{(n-1)/2}<L(2^n)<2^{(3n-3)/4}$
Is anyone able to provide a better and more accurate lower bound than II for $L(2^n)$ to answer (I)?
For instance, when $n=7$, $L(2^7)=11$ because 11 integers less than or equal to $2^7$ are represented by $N: (1, 7, 8, 29, 43, 49, 56, 64, 71, 113, 127)$. It seems to fit my conjectured bound II as $2^3 < L(2^7) < 2^{4.5}$, $8 < 11 < 22.6$.
Thanks for help.