Artin–Hasse exponential

2020 Mathematics Subject Classification: Primary: 11S80 [MSN][ZBL]

A modification of the exponential function in the $p$-adic number domain. In classical analysis we have $$ \exp(z) = \prod_{n=1}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ , $$ which is an identity in formal power series over the rational numbers.

Over the field of $p$-adic numbers we define $$ E_p(z) = \prod_{n=1; p\not\mid n}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ , $$ removing the factors for which $n$ is divisible by $p$. This has radius of convergence $1$ and defines an analytic function with the property that $$ E_p(z) = \exp\left({ z + \frac{z^p}{p} + \frac{z^{p^2}}{p^2} + \cdots }\right) $$ and is given by a power series with rational $p$-integral coefficients.

References

• Cassels, J.W.S. Local fields London Mathematical Society Student Texts 3 Cambridge University Press (1986) ISBN 0-521-31525-5 Zbl 0595.12006
• Robert, Alain M. A course in p-adic analysis Graduate Texts in Mathematics 198 Springer (2000) Zbl 0947.11035
• Schikhof, W.H. Ultrametric calculus. An introduction to p-adic analysis Cambridge Studies in Advanced Mathematics 4 Cambridge University Press (1984) Zbl 0553.26006