On the least quadratic non-residue

Yuk Kam Lau (1) , Jie Wu (2)

1 Department of Mathematics
2 IECN - Institut Élie Cartan de Nancy

We prove that for almost all real primitive characters $\chi_d$ of modulus $|d|$, the least positive integer $n_{\chi_d}$ at which $\chi_d$ takes a value not equal to 0 and 1 satisfies $n_{\chi_d}\ll \log|d|$, and give a quite precise estimate on the size of the exceptional set. Also, we generalize Burgess' bound for $n_{\chi_{p'}}$ (with $p'$ being a prime up to $\pm$ sign) to composite modulus $|d|$ and improve Garaev's upper bound for the least quadratic non-residue in Pajtechi\u\i-\u Sapiro's sequence.

Mots clés: Special sequences

Domaines: Théorie des nombres [math.NT]

Yuk Kam Lau, Jie Wu. On the least quadratic non-residue. International Journal of Number Theory, 2008, 4 (3), pp.423-435. ⟨hal-00097136⟩