Let I(n) denote the number of isomorphism classes of subgroups of (Z/nZ) × , and let G(n) denote the number of subgroups of (Z/nZ) × counted as sets (not up to isomorphism). We prove that both log G(n) and log I(n) satisfy Erdős-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that log G(n) is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of log G(n) and log I(n). 7

Let ω(n) (resp. Ω(n)) denote the number of prime divisors (resp. with multiplicity) of a natural number n. In 1917, Hardy and Ramanujan proved that the normal order of ω(n) is log log n, and the same is true of Ω(n); roughly speaking, a typical natural number n has about log log n prime factors. We prove a similar result for ω(s(n)), where s(n) denotes the sum of the proper divisors of n: For all n ≤ x not belonging to a set of size o(x), |ω(s(n)) − log log s(n)| < ǫ log log s(n), and the same is true for Ω(s(n)).

Abstract. A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer g is a primitive root, provided g = −1 and g is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial g(t) which is not an vth power for any prime v dividing q − 1, there are bounded gaps between monic irreducible polynomials P (t) in F q [t] for which g(t) is a primitive root (which is to say that g(t) generates the group of units modulo P (t)). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice g(t) = t.

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A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer g is a primitive root, provided g = −1 and g is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial g(t) which is not an vth power for any prime v dividing q − 1, there are bounded gaps between monic irreducible polynomials P (t) in F q [t] for which g(t) is a primitive root (which is to say that g(t) generates the group of units modulo P (t)). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice g(t) = t.