Abstract
Let \(K\) be an imaginary quadratic number field and \(\mathcal{O}_K\) be its ring of integers. We show that, if the arithmetic functions \(f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}\) both have level of distribution \(\vartheta\) for some \(0<\vartheta\leq 1/2\) then the Dirichlet convolution \(f*g\) also has level of distribution \(\vartheta\). As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.
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We thank the referee for a detailed report and insightful suggestions that have improved the quality of the manuscript.
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Darbar, P., Mukhopadhyay, A. A Bombieri-type theorem for convolution with application on number field. Acta Math. Hungar. 163, 37–61 (2021). https://doi.org/10.1007/s10474-020-01104-8
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DOI: https://doi.org/10.1007/s10474-020-01104-8
Key words and phrases
- number field
- multiplicative function
- convolution
- distribution of prime ideals
- Titchmarsh divisor problem