Pieter Moree

The invertible residue classes modulo n, (Z/nZ)^*,  form a group of
exponent lambda(n), the Carmichael function. An element g of order
lambda(n) is said to be a primitive lambda-root. In case n is a prime '
we have lambda(n)=n-1 and an element g of that order is said to be a
primitive root.
  Despite the simplicity of the notion of multiplicative order, our
understanding of it is rather poor. I will give a survey on this topic,
with special focus on my own contributions over the years and address
such questions as how often the order is maximal, how often it is even,
and consider equidistribution (or lack thereof) of the order in arithmetic
progressions of small modulus. In most cases we fix g, and let n run
through the prime numbers.