A note on the two variable Artin's conjecture

Abstract

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set$\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}$ has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs $(a,b)$$\#\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}\gg \frac{x}{{\log }^{2}x}.$ In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers $S=\{m_1,m_2,m_3\}$ such that$m_1,m_2,m_3,-3m_1{m}_2,-3m_2{m}_3,-3m_1{m}_3,m_1{m}_2{m}_3$ are not squares, there exists a pair of elements $a,b\in S$ such that$\#\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}\gg \frac{x\log \log x}{{\log }^{2}x}.$ Further, under the assumption of a level of distribution greater than $x^{\frac{2}{3}}$ in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers $S=\{m_1,m_2\}$ such that$m_1,m_2,-3m_1{m}_2$ are not squares, there exists a pair of elements $a,b\in \{m_1,m_2,-3m_1{m}_2\}$ such that$\#\{p⩽x:p\text{ prime, }\phantom{m}a\mathrm{mod}p\in 〈b〉\mathrm{mod}p\}\gg \frac{x\log \log x}{{\log }^{2}x}.$