Abstract
We give an asymptotic formula for \(\sum _{1\le n_1.n_2, \ldots ,n_l\le x^{1/r}}\tau _k(n^r_1+n^r_2+\ldots +n^r_l)\), where \(\tau _k(n)\) represents the k-th divisor function. Previously, partial results of this summation were obtained by many scholars.
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Acknowledgements
The authors would like to thank the editor and referee for their careful reading the manuscript. We thank the referee for his/her valuable suggestions which significantly improved the quality of our paper. The authors would also like to thank Prof. Liyu Liu, Dr. Guilin Li, Dr. Yue-Feng She, and Dr. Li-Yuan Wang for their helpful comments.
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The second author was supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20210784). He was also supported by the foundations of the projects “Jiangsu Provincial Double–Innovation Doctor Program” (No. JSSCBS20211023) and “Golden Phenix of the Green City–Yang Zhou” to excellent PhD (No. YZLYJF2020PHD051)
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Zhou, GL., Ding, Y. Sums of the higher divisor function of diagonal homogeneous forms. Ramanujan J 59, 933–945 (2022). https://doi.org/10.1007/s11139-022-00579-z
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DOI: https://doi.org/10.1007/s11139-022-00579-z