For any real number $ x, [x] $ denotes the integer part of $ x $. $ \mathcal{F}_{1} $ denotes a class of multiplicative functions that are small in a numerical sense. In this paper, we study the distribution of the functions from $ \mathcal{F}_{1} $ in Piatetski-Shapiro sequences. In particular, we prove that for $ 1 < c < \frac{4}{3}, $
$ \sum\limits_{\stackrel{n\leq x}{f\in \mathcal{F}_{1}}} f([n^{c}]) = \int_{1}^{x^{c}} \gamma u^{\gamma-1} d\left( \sum\limits_{1\leq n\leq u} f(n)\right) +O(x^{1-\varepsilon}), $
where $ \gamma = \frac{1}{c} $ and $ [n^{c}] $ is the Piatetski-Shapiro sequence.
Citation: Haihong Fan. A class of small multiplicative functions in Piatetski-Shapiro sequences[J]. AIMS Mathematics, 2025, 10(12): 28514-28523. doi: 10.3934/math.20251255
For any real number $ x, [x] $ denotes the integer part of $ x $. $ \mathcal{F}_{1} $ denotes a class of multiplicative functions that are small in a numerical sense. In this paper, we study the distribution of the functions from $ \mathcal{F}_{1} $ in Piatetski-Shapiro sequences. In particular, we prove that for $ 1 < c < \frac{4}{3}, $
$ \sum\limits_{\stackrel{n\leq x}{f\in \mathcal{F}_{1}}} f([n^{c}]) = \int_{1}^{x^{c}} \gamma u^{\gamma-1} d\left( \sum\limits_{1\leq n\leq u} f(n)\right) +O(x^{1-\varepsilon}), $
where $ \gamma = \frac{1}{c} $ and $ [n^{c}] $ is the Piatetski-Shapiro sequence.
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Haihong Fan. A class of small multiplicative functions in Piatetski-Shapiro sequences[J]. AIMS Mathematics, 2025, 10(12): 28514-28523. doi: 10.3934/math.20251255
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