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

Maxie Dion Schmidt

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 $n$.

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

Abstract

Sign changes in sums of arithmetic functions and their inverses are a subtle topic with room to grow new results. Suppose that is the summatory function of some arithmetic function such that . There are known lower bounds on the limiting growth of -- the number of sign changes of on the interval as . We observe a partition theoretic sign smoothing by discrete convolution of the local oscillatory properties of the Dirichlet inverse of , . These so-called invertible ``magic partition function`` encodings lead to a sequence of convolution sums which have predictable sign properties provided the sequence of (, respectively) has reasonable asymptotic upper bounds with respect to .
Paper Structure (7 sections, 2 theorems, 18 equations, 6 tables)

This paper contains 7 sections, 2 theorems, 18 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Dirichlet convolutions and Dirichlet inverse functions
  3. Some curious convolution experiments with partition functions
  4. Local sign changes of an arithmetic function
  5. Proof of the main theorem
  6. Conclusions
  7. Tables of experiments

Key Result

Theorem 1.4

Let the sign function, $\mathop{\mathrm{sgn}}\nolimits(h(n)) \mapsto \{\pm 1\}$, be defined to be $|h(n)| / h(n)$ whenever $h(n) \neq 0$ where $\mathop{\mathrm{sgn}}\nolimits(h(n)) := 1$ if $h(n) = 0$. Furthermore, suppose that $f(n) \geq 1$ for all $n$ and that $f(n) \ll q^{\ast}(n)$ as $n \rightar is eventually constant for all sufficiently large $n$.

Theorems & Definitions (6)

  • Definition 1.1
  • Remark 1.2: Asymptotics of the special partition functions
  • Definition 1.3: Convolution-based encodings
  • Theorem 1.4: Sign smoothing convolutions
  • Theorem 1.5: Pólya
  • proof : Proof of Theorem \ref{['theorem_MagicPartitionsObs_v1']}(A)