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 $G(n,n)\neq 0$, and for each $\beta$ define a sequence $(a_k)$ by $\sum_{k\le n} a_k G(n,k)=n^{-\beta}$, solved rank by rank with nothing to assume and no convergence to establish. Then $G$ is regular when the partial sums $\sum_{k\le n} a_k$ change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index $\alpha(G)$, 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 $G(n,k)=(k/n)\lfloor n/k\rfloor$ 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.