A generalization to number fields of Euler's theorem on the series of reciprocals of primes

2019/11/09 by Tringali, Salvatore
#11R04 #11R27 (Primary) #11R42 #13F15 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.03732

Abstract

Let X be a set of positive integers, and let \mathbb Z_K be the ring of integers of a number field K of degree n. Denote by N(I) the absolute norm of an ideal I of \mathbb Z_K, and by \mathcal A the set of principal ideals a\mathbb Z_K such that a is an atom of \mathbb Z_K and a divides m for some m ∈ X. Building upon the ideas of Clarkson from [Proc. Amer. Math. Soc. 17 (1966), 541], we show that, if the series ∑_{m ∈ X} 1/m diverges, then so does the series ∑_{\mathfrak a ∈ \mathcal A} |N(\mathfrak a)|^-1/n. Most notably, this generalizes a classical theorem of Euler on the series of reciprocals of positive rational primes.

Related

• Sum-product phenomenon in quotients of rings of algebraic integers
• Bertrand's Postulate for Number Fields
• Euler constants from primes in arithmetic progression
• Euler's factorial series at algebraic integer points
• Sumsets of reciprocals in prime fields and multilinear Kloosterman sums