Artin-Hasse exponential
From Wikipedia, the free encyclopedia
In mathematics, the Artin-Hasse exponential is the power series given by
[edit] Properties
- The coefficients are p-integral; in other words, their denominators are not divisible by p. This follows from Dwork's lemma, which says that a power series f(x) = 1+... with rational coefficients has p-integral coefficients if and only if f(xp)/f(x)p ≡ 1 mod p.
- The coefficient of xn of n!E_p(x) is the number of elements of the symmetric group on n points of order a power of p. (This gives another proof that the coefficients are p-integral, using the fact that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d.)
- It can be written as the infinite product
-
-
Ep(x) = ∏ (1 − xn)μ(n) / n (p,n) = 1
-
(The function μ is the Möbius function.)
[edit] See also
[edit] References
- A course in p-adic analysis, by Alain M. Robert
 |
This number theory-related article is a stub. You can help Wikipedia by expanding it. |
