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The irrationality of a divisor function series of Erdős and Kac

Volume 211 / 2023

Kyle Pratt Acta Arithmetica 211 (2023), 193-228 MSC: Primary 11J72. DOI: 10.4064/aa220927-1-9 Published online: 8 November 2023

Abstract

For positive integers and n let \sigma _k(n) denote the sum of the kth powers of the divisors of n. Erdős and Kac asked whether, for every k, the number \alpha _k = \sum _{n\geq 1} \frac {\sigma _k(n)}{n!} is irrational. It is known unconditionally that \alpha _k is irrational if k\leq 3. We prove that \alpha _4 is irrational.

Authors

  • Kyle PrattDepartment of Mathematics
    Brigham Young University
    Provo, UT 84604, USA
    e-mail

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