The irrationality of a divisor function series of Erdős and Kac

2022/09/22 by Kyle Pratt, Pratt, Kyle
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.11124

openalex publication_date 2022/09/22 · openalex created_date 2022/09/25 · openalex updated_date 2026/07/28

Abstract

For positive integers k and n let σ_k(n) denote the sum of the kth powers of the divisors of n. Erdős and Kac asked whether, for every k, the number α_k = ∑_{n≥ 1} (σ_k(n))/(n!) is irrational. It is known unconditionally that α_k is irrational if k≤ 3. We prove α₄ is irrational.

Related

• Irrationality of the sum of a p-adic series
• On several irrationality problems for Ahmes series
• The irrationality of a prime factor series under a prime tuples conjecture
• On an incomplete argument of Erdos on the irrationality of Lambert series
• General proof for irrationality of infinite sums based on Fourier's proof