Artin–Hasse exponential
In mathematics, the Artin–Hasse exponential, named after Emil Artin and Helmut Hasse, is the power series given by
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! Ep(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
-
- (The function μ is the Möbius function.) This resembles the exponential series, in the sense that taking this product over all n rather than only n prime to p is an infinite product which converges (in the ring of formal power series) to the exponential series.
See also
References
- A course in p-adic analysis, by Alain M. Robert
- Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 9780821832592, MR1915966