Abstract
For each \(s\in {\mathbb {R}}\) and \(n\in {\mathbb {N}}\), let \(\sigma _s(n) = \sum _{d\mid n}d^s\). In this article, we study the number of sign changes in the difference \(\sigma _s(an+b)-\sigma _s(cn+d)\) where a, b, c, d, s are fixed, the vectors (a, b) and (c, d) are linearly independent over \({\mathbb {Q}}\), and n runs over all positive integers. We also give several examples and propose some problems.
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Acknowledgements
Prapanpong Pongsriiam’s research project is jointly funded by the Faculty of Science Silpakorn University and the National Research Council of Thailand (NRCT), Grant Number NRCT5-RSA63021-02. The author is also supported by the Tosio Kato Fellowship given by the Mathematical Society of Japan during his visit at Nagoya University in July 2022 to July 2023. Napp Phunphayap helped him with the computer programming. The anonymous reviewer gave him many kind and helpful comments which improve the quality of this article. He is grateful to them all.
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Pongsriiam, P. Sums of divisors on arithmetic progressions. Period Math Hung (2023). https://doi.org/10.1007/s10998-023-00566-x
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DOI: https://doi.org/10.1007/s10998-023-00566-x