On Equal Consecutive Values of Multiplicative Functions
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917827846340608 |
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| author | Mangerel, Alexander P. |
| author_facet | Mangerel, Alexander P. |
| contents | Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) = f(n+h) \neq 0\} $$ has logarithmic density 0. We also prove a converse result, along with an application to the Fourier coefficients of holomorphic cusp forms.
The proof involves analysing the value distribution of $f$ using the compositions $|f|^{it}$, relying crucially on various applications of Tao's theorem on logarithmically-averaged correlations of non-pretentious multiplicative functions. Further key inputs arise from the inverse theory of sumsets in continuous additive combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09929 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Equal Consecutive Values of Multiplicative Functions Mangerel, Alexander P. Number Theory Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) = f(n+h) \neq 0\} $$ has logarithmic density 0. We also prove a converse result, along with an application to the Fourier coefficients of holomorphic cusp forms. The proof involves analysing the value distribution of $f$ using the compositions $|f|^{it}$, relying crucially on various applications of Tao's theorem on logarithmically-averaged correlations of non-pretentious multiplicative functions. Further key inputs arise from the inverse theory of sumsets in continuous additive combinatorics. |
| title | On Equal Consecutive Values of Multiplicative Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.09929 |