Regular Arithmetic Functions, Volume I. Theory, Applications, Examples
Abstract
This is the first of two volumes on regular arithmetic functions, and an introduction to their theory. A regular arithmetic function (RAF) is a kernel whose properties come from a single defining equation. Let G(n,k) be a function of two integer variables with G(n,n)≠ 0, and for each β define a sequence (ak) by ∑k≤ n ak G(n,k)=n−β, solved rank by rank with nothing to assume and no convergence to establish. Then G is regular when the partial sums ∑k≤ n ak change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index α(G), a quantity belonging to the kernel itself. The index came out of analogies, experiment and observation, and it is arithmetic by nature. Its most visible application is the Riemann hypothesis, which holds if and only if Ingham's kernel G(n,k)=(k/n)⌊ n/k⌋ is a RAF of index 1/2. It is not the only one. On a problem of a quite different kind the same theory improves the known decay bound for the orthorecursive expansion of unity. Seventeen kernels are worked out in a gallery, so that the notion can be handled by the reader. What is conjectural, conditional or open is marked as such and collected in a register. Volume II is given over to that kernel alone, with its connection to the world of Hasse-Weil zeta functions through a gauged system.
Related
- Geometric families of multiple elliptic Gamma functions and arithmetic applications, III
- Sharp Volume-Based Bounds for BMO Functions on Compact Domains and Applications
- Examples of Bessel Functions of Order Zero Related to the Functional Equation of the Riemann Zeta Function
- Triple divisor type functions in arithmetic progressions to prime power moduli
- Truncated pretentious distances of multiplicative arithmetic functions
BibTeX
@misc{cloitre2026regular,
title = {{Regular Arithmetic Functions, Volume I. Theory, Applications, Examples}},
author = {Benoit Cloitre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09366},
eprint = {2609.09366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09366},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regular Arithmetic Functions, Volume I. Theory, Applications, Examples
AU - Benoit Cloitre
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.09366
UR - https://arxiv.org/abs/2609.09366
AB - This is the first of two volumes on regular arithmetic functions, and an introduction to their theory. A regular arithmetic function (RAF) is a kernel whose properties come from a single defining equation. Let G(n,k) be a function of two integer variables with G(n,n)≠ 0, and for each β define a sequence (a_k) by ∑_k≤ n a_k G(n,k)=n^−β, solved rank by rank with nothing to assume and no convergence to establish. Then G is regular when the partial sums ∑_k≤ n a_k change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index α(G), a quantity belonging to the kernel itself. The index came out of analogies, experiment and observation, and it is arithmetic by nature. Its most visible application is the Riemann hypothesis, which holds if and only if Ingham's kernel G(n,k)=(k/n)⌊ n/k⌋ is a RAF of index 1/2. It is not the only one. On a problem of a quite different kind the same theory improves the known decay bound for the orthorecursive expansion of unity. Seventeen kernels are worked out in a gallery, so that the notion can be handled by the reader. What is conjectural, conditional or open is marked as such and collected in a register. Volume II is given over to that kernel alone, with its connection to the world of Hasse-Weil zeta functions through a gauged system.
ER -