The Artin–Hasse exponential is the generating function for the probability a uniformly randomly selected element of Sn (the symmetric group with n elements) has p-power order (the number of which is denoted by tp,n): This gives a third proof that the coefficients of Ep(x) are p-integral, using the theorem of Frobenius that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d. Apply this theorem to the nth symmetric group with d equal to the highest power of p dividing n!.
(1) If G is the p-adic integers, it has exactly one open subgroup of each p-power index and a continuous homomorphism from G to Sn is essentially the same thing as choosing an element of p-power order in Sn, so we have recovered the above combinatorial interpretation of the Taylor coefficients in the Artin–Hasse exponential series.
The special case when G is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and takes the form where am,n is the number of solutions to gm = 1 in Sn.
[citation needed] At the 2002 PROMYS program, Keith Conrad conjectured that the coefficients of
are uniformly distributed in the p-adic integers with respect to the normalized Haar measure, with supporting computational evidence.
Dinesh Thakur has also posed the problem of whether the Artin–Hasse exponential reduced mod p is transcendental over