Magic partition functions: Sign smoothing convolutions with Dirichlet invertible arithmetic functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914377645424640 |
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| author | Schmidt, Maxie Dion |
| author_facet | Schmidt, Maxie Dion |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06890 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Magic partition functions: Sign smoothing convolutions with Dirichlet invertible arithmetic functions Schmidt, Maxie Dion Number Theory 11A25, 11N64, 11N56 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 $n$. |
| title | Magic partition functions: Sign smoothing convolutions with Dirichlet invertible arithmetic functions |
| topic | Number Theory 11A25, 11N64, 11N56 |
| url | https://arxiv.org/abs/2603.06890 |