On the Hyperbolic Fractional Sum of the Divisor Function
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866913054028988416 |
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| author | Li, Ling |
| author_facet | Li, Ling |
| contents | Let $τ(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} τ\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} τ\left( \left[ \frac{x}{n} \right] \right) τ(n), $$ where $[t]$ denotes the integral part of the real number $t$. By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, we obtain an improved asymptotic formula for $T(x)$. In particular, we show that for any $\varepsilon > 0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\varepsilon})$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\varepsilon$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20400 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Hyperbolic Fractional Sum of the Divisor Function Li, Ling Number Theory 11L07, 11N37, 11A25, 11D75 Let $τ(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} τ\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} τ\left( \left[ \frac{x}{n} \right] \right) τ(n), $$ where $[t]$ denotes the integral part of the real number $t$. By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, we obtain an improved asymptotic formula for $T(x)$. In particular, we show that for any $\varepsilon > 0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\varepsilon})$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\varepsilon$. |
| title | On the Hyperbolic Fractional Sum of the Divisor Function |
| topic | Number Theory 11L07, 11N37, 11A25, 11D75 |
| url | https://arxiv.org/abs/2604.20400 |