abstract

E-functions form a class of entire functions given by power series with algebraic coefficients, introduced by Siegel in 1929. Motivation for their study is provided by classical examples such as the exponential function and the Bessel function. Their arithmetic properties are particularly striking: a result of Siegel and Shidlovskii on their special values, 'E-values', allows for simple transcendence proofs and constitutes a vast generalisation of the Lindemann-Weierstrass theorem. E-values are closely linked to a class of complex numbers called exponential periods, which are complex numbers given by particular integrals over varieties involving the exponential function. The interplay between these E-values and exponential periods is an active area of research and we will discuss a couple of results in this direction.