アルティン・ハッセの指数関数
x n , {\displaystyle \exp(\sum _{d|m}x^{d}/d)=\sum _{n\geq 0}{\frac {a_{m,n}}{n!}}x^{n},}
where am,n is the number of solutions to g = 1 in Sn.
David Roberts provided a natural combinatorial link between the Artin–Hasse exponential and the regular exponential in the spirit of the ergodic perspective (linking the p-adic and regular norms over the rationals) by showing that the Artin–Hasse exponential is also the generating function for the probability that an element of the symmetric group is unipotent in characteristic p, whereas the regular exponential is the probability that an element of the same group is unipotent in characteristic zero.
At the 2002 PROMYS program, Keith Conrad conjectured that the coefficients of E p ( x ) {\displaystyle E_{p}(x)} are uniformly distributed in the p-adic integers with respect to the normalized Haar measure, with supporting computational evidence. The problem is still open.
Dinesh Thakur has also posed the problem of whether the Artin–Hasse exponential reduced mod p is transcendental over F p ( x ) {\displaystyle \mathbb {F} _{p}(x)} .