Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $\epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.Comment: to appear in Bulletin of the Australian Mathematical Societ

Set Equidistribution of subsets of (Z/nZ) *

In 2010, Murty and Thangadurai [MuTh10] provided a criterion for the set equidistribution of residue classes of subgroups in (Z/nZ) *. In this article, using similar methods, we study set equidistribution for some class of subsets of (Z/nZ) *. In particular, we study the set equidistribution modulo 1 of cosets, complement of subgroups of the cyclic group (Z/nZ) * and the subset of elements of fixed order, whenever the size of the subset is sufficiently large