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Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers
- Source :
- Journal of Number Theory, Vol. 265, 138--180 (2024)
- Publication Year :
- 2024
-
Abstract
- For all $\alpha_1,\alpha_2\in(1,2)$ with $1/\alpha_1+1/\alpha_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor{n_1^{\alpha_1}}\rfloor+\lfloor{n_2^{\alpha_2}}\rfloor$ is equal to $\Gamma(1+1/\alpha_1)\Gamma(1+1/\alpha_2)\Gamma(1/\alpha_1+1/\alpha_2)^{-1}N^{1/\alpha_1+1/\alpha_2-1} + o(N^{1/\alpha_1+1/\alpha_2-1})$ as $N\to\infty$, where $\Gamma$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when $\alpha_1,\alpha_2\in(1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.<br />Comment: 44 pages, 4 figures
Details
- Database :
- arXiv
- Journal :
- Journal of Number Theory, Vol. 265, 138--180 (2024)
- Publication Type :
- Report
- Accession number :
- edsarx.2403.16691
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jnt.2024.06.012