Magic partition functions: Sign smoothing convolutions with Dirichlet invertible arithmetic functions

TL;DR

This paper introduces a novel approach using Dirichlet invertible arithmetic functions and convolution sums to analyze and predict sign changes in sums of arithmetic functions, advancing understanding of their oscillatory behavior.

Contribution

It proposes a new partition theoretic sign smoothing method via convolution with Dirichlet inverses, providing predictable sign properties under certain asymptotic bounds.

Findings

Sign change behavior can be controlled through convolution with Dirichlet inverses.

Predictable sign properties depend on asymptotic bounds of the sequences.

Introduces the concept of invertible 'magic partition functions' for sign analysis.

Abstract

Sign changes in sums of arithmetic functions and their inverses are a subtle topic with room to grow new results. Suppose that $S_f(x) := \sum_{n \leq x} f(n)$ is the summatory function of some arithmetic function $f$ such that $f(1) \neq 1$. There are known lower bounds on the limiting growth of $V(S_f, Y)$ -- the number of sign changes of $S_f(y)$ on the interval $y \in (0, Y]$ as $Y \rightarrow \infty$. We observe a partition theoretic sign smoothing by discrete convolution of the local oscillatory properties of the Dirichlet inverse of $f$, $S_{f^{-1}}(x)$. These so-called invertible ``magic partition function`` encodings lead to a sequence of convolution sums which have predictable sign properties provided the sequence of $f(n)$ ($f^{-1}(n)$, respectively) has reasonable asymptotic upper bounds with respect to $n$.