Artin–Hasse exponential

In mathematics, the Artin–Hasse exponential, introduced by , is the power series given by

One motivation for considering this series to be analogous to the exponential function comes from infinite products. In the ring of formal power series Q[[x]] we have the identity

where μ(n) is the Möbius function. This identity can be verified by showing the logarithmic derivative of the two sides are equal and that both sides have the same constant term. In a similar way, one can verify a product expansion for the Artin–Hasse exponential:

So passing from a product over all n to a product over only n prime to p, which is a typical operation in p adic analysis, leads from ex to Ep(x).

The coefficients of Ep(x) are rational. We can use either formula for Ep(x) to prove that, unlike ex, all of its coefficients are p integral; in other words, the denominators of the coefficients of Ep(x) are not divisible by p. A first proof uses the definition of Ep(x) and 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 pZp[[x]]. When f(x) = Ep(x), we have f(xp)/f(x)p = e−px, whose constant term is 1 and all higher coefficients are in pZp. A second proof comes from the infinite product for Ep(x): each exponent μ(n)/n for n not divisible by p is a p integral, and when a rational number a is p integral all coefficients in the binomial expansion of (1 xn)a are p integral by p adic continuity of the binomial coefficient polynomials t(t 1) (t k+1)/k! in t together with their obvious integrality when t is a nonnegative integer (a is a p adic limit of nonnegative integers) Thus each factor in the product of Ep(x) has p integral coefficients, so Ep(x) itself has p integral coefficients. The (p integral) series expansion has radius of convergence 1.

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. The problem is still open. Dinesh Thakur has also posed the problem of whether the Artin–Hasse exponential reduced mod p is transcendental over .