On a certain arithmetic function defined via Bernoulli numbers
Abstract
The paper defines a certain function χ(n) on the set of odd integers n ≥ 3. The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if n is a prime number or an odd Giuga number. Thus, the integrality of χ(n) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt--Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers p for which χ(p)=p−1 and Novák--Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes Pμ of primes with a fixed value χ(p)=μ, a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number Bp−1. For some μ, explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.
Related
- Inequalities involving the Riemann zeta function and implications for Bernoulli numbers
- Some Expressions involving Bernoulli, Harmonic, and Stirling Numbers
- Two Legendre-symbol determinants of Sun and Bernoulli numbers
- Partial Results on Hankel Determinants and the Corresponding J-Fractions of a Sequence Related to Bernoulli Numbers
- Identities for generalized Bernoulli numbers from a Lie model of the 2-disk
BibTeX
@misc{svinin2026certain,
title = {{On a certain arithmetic function defined via Bernoulli numbers}},
author = {Andrei K. Svinin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.18607},
eprint = {2607.18607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18607},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a certain arithmetic function defined via Bernoulli numbers
AU - Andrei K. Svinin
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.18607
UR - https://arxiv.org/abs/2607.18607
AB - The paper defines a certain function χ(n) on the set of odd integers n ≥ 3. The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if n is a prime number or an odd Giuga number. Thus, the integrality of χ(n) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt--Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers p for which χ(p)=p−1 and Novák--Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes P_μ of primes with a fixed value χ(p)=μ, a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number B_p−1. For some μ, explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.
ER -