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| In mathematics, the '''Artin–Hasse exponential''', named after [[Emil Artin]] and [[Helmut Hasse]], is the [[power series]] given by
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| :<math> E_p(x) = \exp\left(x + \frac{x^p}{p} + \frac{x^{p^2}}{p^2} + \frac{x^{p^3}}{p^3} +\cdots\right).</math>
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| ==Motivation==
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| Unlike the traditional exponential series, 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''(''x''<sup>''p''</sup>)/''f''(''x'')<sup>''p''</sup> ≡ 1 mod ''p''.
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| Following this definition, by [[Möbius inversion]] it can be written as the infinite product
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| ::<math>E_p(x) = \prod_{(p,n)=1}(1-x^n)^{-\mu(n)/n}. \,</math>
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| : (The function μ is the [[Möbius function]].) This follows from the analogous relation for the regular 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.
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| ==Combinatorial interpretation==
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| The Artin–Hasse exponential is the [[generating function]] for the probability a uniformly randomly selected element of ''S''<sub>''n''</sub> (the [[symmetric group]] with ''n'' elements) has ''p''-power order (the number of which is denoted by ''t''<sub>''p,n''</sub>):
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| :<math>E_p(x)=\sum_{n\ge 0} \frac{t_{p,n}}{n!}x^n.</math>
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| 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''.
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| More generally, for any topologically finitely generated profinite group ''G'' there is an identity
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| :<math>\exp(\sum_{H \subset G} x^{[G:H]}/[G:H])=\sum_{n\ge 0} \frac{a_{G,n}}{n!}x^n,</math>
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| where ''H'' runs over open subgroups of ''G'' with finite index (there are finitely many of each index since ''G'' is topologically finitely generated) and ''a<sub>G,n</sub>'' is the number of continuous homomorphisms from ''G'' to ''S<sub>n</sub>''. Two special cases are worth noting. (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 ''S<sub>n</sub>'' is essentially the same thing as choosing an element of ''p''-power order in ''S<sub>n</sub>'', so we have recovered the above combinatorial interpretation of the Taylor coefficients in the Artin–Hasse exponential series. (2) If ''G'' is a finite group then the sum in the exponential is a finite sum running over all subgroups of ''G'', and continuous homomorphisms from ''G'' to ''S<sub>n</sub>'' are simply homomorphisms from ''G'' to ''S<sub>n</sub>''. The result in this case is due to Wohlfahrt (1977). The special case when ''G'' is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and takes the form
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| :<math>\exp(\sum_{d|m} x^d/d)=\sum_{n\ge 0} \frac{a_{m,n}}{n!}x^n,</math> | |
| where ''a<sub>m,n</sub>'' is the number of solutions to ''g<sup>m</sup>'' = 1 in ''S<sub>n</sub>''.
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| [[David Roberts]]{{dn|date=July 2013}} 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 (algebra)|characteristic]] ''p'', whereas the regular exponential is the probability that an element of the same group is unipotent in characteristic zero.
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| ==Conjectures==
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| At the 2002 [[PROMYS]] program, [[Keith Conrad]] conjectured that the coefficients of <math>E_p(x)</math> are uniformly distributed in the [[p-adic number|p-adic]] integers with respect to the normalized Haar measure, with supporting computational evidence. The problem is still open.
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| Dinesh Thakur has also posed the problem of whether the Artin–Hasse exponential reduced mod ''p'' is transcendental over <math>\mathbb{F}_p(x)</math>.
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| Various other relatively simple properties of the functions are also open, including whether it satisfies the traditional exponential functional equation <math>E_p(a)^b=E_p(ab)</math> and analogous relations defined for a general exponential by utilizing its inverse, the Artin–Hasse logarithm.
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| ==See also==
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| *[[Witt vector]]
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| *[[Formal group]]
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| ==References==
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| *''A course in p-adic analysis'', by Alain M. Robert
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| * {{Citation
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| | last=Fesenko
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| | first=Ivan B.
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| | last2=Vostokov
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| | first2=Sergei V.
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| | title=Local fields and their extensions
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| | publisher=[[American Mathematical Society]]
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| | location=Providence, RI
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| | year=2002
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| | series=Translations of Mathematical Monographs
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| | volume=121
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| | edition=Second
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| | isbn=978-0-8218-3259-2
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| | mr=1915966
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| }}
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| {{DEFAULTSORT:Artin-Hasse exponential}}
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| [[Category:Number theory]]
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