Quadratic residue patterns of length 4 and 5

Abstract

We generalize the results of the first part of the paper by Kiritchenko, Tsfasman, Vlăduţ and Zakharevich on quadratic residue patterns to the case of arbitrary words over the alphabet $\{R,N\}$ of length $l=4$ and $l=5$. First, we obtain explicit formulas for the number of occurrences of a given pattern $S$ in the sequence of quadratic residues and nonresidues modulo a prime $p$. These formulas are expressed in terms of Frobenius traces on a finite collection of low-genus hyperelliptic curves. Second, using a generalized Sato-Tate approach, we describe the limiting distributions of the normalized error term for $l=4$.

17 pages, 6 tables