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Beatty sequences and prime numbers with restrictions on strongly q-additive functions

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Abstract

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes \(p=\lfloor \alpha n +\beta \rfloor \) and \(f(p)\equiv a (\mathrm{mod\,}b)\), with \(n\ge N \), where \(\alpha \), \(\beta \) are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer \(k\ge 3 \), all sufficiently large \(N\equiv k (\mathrm{mod\,}2) \) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.

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Acknowledgments

The author thanks the referee for many detailed comments and suggestions.

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  1. Faculté des Sciences de Sfax, BP 1171, 3000, Sfax, Tunisia

    Mohamed Mkaouar

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  1. Mohamed Mkaouar
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Correspondence to Mohamed Mkaouar.

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Mkaouar, M. Beatty sequences and prime numbers with restrictions on strongly q-additive functions. Period Math Hung 72, 139–150 (2016). https://doi.org/10.1007/s10998-016-0114-7

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