We present a completely explicit transcendence measure for e. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of e. The results are based on Hermite-Padé approximations and on careful analysis of common factors in the footsteps of Hata.

We study a linear form in the values of Euler's series F (t) = ∞ n=0 n!t n at algebraic integer points α 1 , . . . , α m ∈ Z K belonging to a number field K. Let v|p be a non-Archimedean valuation of K. Two types of non-vanishing results for the linear form Λare derived, the second of them containing a lower bound for the v-adic absolute value of Λ v . The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series ∞ n=0 n−1 k=0 P (k) t n , where P (x) is a polynomial of degree one.

We study a linear form in the values of Euler's series F (t) = ∞ n=0 n!t n at algebraic integer points α 1 , . . . , α m ∈ Z K belonging to a number field K. Let v|p be a non-Archimedean valuation of K. Two types of non-vanishing results for the linear form Λare derived, the second of them containing a lower bound for the v-adic absolute value of Λ v . The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series ∞ n=0 n−1 k=0 P (k) t n , where P (x) is a polynomial of degree one.

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