On the Function Field Analogue of Landau's Theorem on Sums of Squares

Lior Bary-Soroker; Yotam Smilansky; Adva Wolf

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arXiv · 1504.06809

On the Function Field Analogue of Landau's Theorem on Sums of Squares

Lior Bary-Soroker·Yotam Smilansky·Adva Wolf

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Abstract

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in $\mathbb{Z}$. We define the analogue of a sum of two squares in $\mathbb{F}_q[T]$ and estimate the number $B_q(n)$ of such polynomials of degree $n$ in two cases. The first case is when $q$ is large and $n$ fixed and the second case is when $n$ is large and $q$ is fixed. Although the methods used and main terms computed in each of the two cases differ, the two iterated limits of (a normalization of) $B_q(n)$ turn out to be exactly the same.

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BibTeX RIS

Lior Bary-Soroker, Yotam Smilansky, Adva Wolf. 2015-04-26. On the Function Field Analogue of Landau's Theorem on Sums of Squares. https://arxiv.org/abs/1504.06809

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