Euler's factorial series at algebraic integer points

2018/09/28 by Seppälä, Louna
#11J61 #41A21 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.10997

Abstract

We study a linear form in the values of Euler's series F(t)=∑_n=0^∞ n!tⁿ at algebraic integer points α₁, …, α_m ∈ ℤ_\mathbbK belonging to a number field \mathbbK. Let v|p be a non-Archimedean valuation of \mathbbK. Two types of non-vanishing results for the linear form Λ_v = λ₀ + λ₁ F_v(α₁) + … + λ_m F_v(α_m), λ_i ∈ ℤ_\mathbbK, are 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^∞ ( ∏_k=0^n-1 P(k) ) tⁿ, where P(x) is a polynomial of degree one.

Related

• Explicit results for Euler's factorial series in arithmetic progressions under GRH
• Euler's factorial series, Hardy integral, and continued fractions
• Euler's divergent series in arithmetic progressions
• A generalization to number fields of Euler's theorem on the series of reciprocals of primes
• Euler constants from primes in arithmetic progression