Geodesic
On Equal Consecutive Values of Multiplicative Functions
Discrete analysis (2024)

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arXiv
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.
Publié le : 
Alexander P. Mangerel. On Equal Consecutive Values of Multiplicative Functions. Discrete analysis (2024). http://geodesic.mathdoc.fr/item/DAS_2024_a9/
@article{DAS_2024_a9,
     author = {Alexander P. Mangerel},
     title = {On {Equal} {Consecutive} {Values} of {Multiplicative} {Functions}},
     journal = {Discrete analysis},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2024_a9/}
}
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