Landau's theorem asserts that the asymptotic density of sums of two squares in the interval $1\leq n\leq x$ is $K/{\sqrt{\log x}}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $|n-x|\leq x^{\epsilon}$ for a fixed $\epsilon$ and $x\to\infty$. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic $f_0\in\Bbb{F}_q[T]$ of degree $n$ and take $\epsilon$ with $1>\epsilon\geq {2\over n}$. Then the asymptotic density of polynomials $f$ in the "interval" $\deg(f-f_0)\leq\epsilon n$ that are of the form $f=A^2+TB^2$, $A,B\in\Bbb{F}_q[T]$ is ${1\over 4^n}\big(\matrix{2n\atop n}\big)$ as $q\to\infty$. This density agrees with the asymptotic density of such monic $f$'s of degree $n$ as $q\to\infty$, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of $f(-T^2)$, where $f$ is a polynomial of degree $n$ with a few variable coefficients: The Galois group is the hyperoctahedral group of order $2^nn!$.


Additional Information

Print ISSN
pp. 1113-1131
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.