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SUMMARY:Paul Péringuey (University of British Columbia)
DTSTART:20241205T210000Z
DTEND:20241205T220000Z
DTSTAMP:20260423T021339Z
UID:ANTS/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ANTS/6/">Ref
 inements of Artin's primitive root conjecture</a>\nby Paul Péringuey (Uni
 versity of British Columbia) as part of Calgary Algebra and Number Theory 
 Seminar\n\nLecture held in MS 337.\n\nAbstract\nLet $\\rm{ord}_p(a)$ be th
 e order of $a$ in $\\left(\\mathbb{Z}/p\\mathbb{Z}\n\\right)^*$. In 1927\,
  Artin conjectured that the set of primes $p$ for which an\ninteger $a\\ne
 q -1\,\\square$ is a primitive root (i.e. $\\rm{ord}_p(a)=p-1$) has\na pos
 itive asymptotic density among all primes. In 1967 Hooley proved this\ncon
 jecture assuming the Generalized Riemann Hypothesis (GRH).\n\nIn this talk
  we will study the behaviour of $\\rm{ord}_p(a)$ as $p$ varies over\nprime
 s\, in particular we will show\, under GRH\, that the set of primes $p$ fo
 r\nwhich $\\rm{ord}_p(a)$ is ``$k$ prime factors away'' from $p-1$ has a p
 ositive\nasymptotic density among all primes except for particular values 
 of $a$ and\n$k$. We will interpret being ``$k$ prime factors away'' in thr
 ee different\nways\, namely $k=\\omega(\\frac{p-1}{\\rm{ord}_p(a)})$\, $k=
 \\Omega(\\frac{p-1}\n{\\rm{ord}_p(a)})$ and $k=\\omega(p-1)-\\omega(\\rm{o
 rd}_p(a))$\, and present\nconditional results analogous to Hooley's in all
  three cases and for all\ninteger $k$. From this\, we will derive conditio
 nally the expectation for these\nquantities.\n\nFurthermore we will provid
 e partial unconditional answers to some of these\nquestions.\n\nThis is jo
 int work with Leo Goldmakher and Greg Martin.\n
LOCATION:https://researchseminars.org/talk/ANTS/6/
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