The second moment of Ramanujan sums
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912826740703232 |
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| author | Ziwei, Hong Zhiyong, Zheng |
| author_facet | Ziwei, Hong Zhiyong, Zheng |
| contents | In this paper, we study $C(x, y)$, the second moment of Ramanujan sums. Assuming the Riemann Hypothesis(RH), we establish an asymptotic formula for $C(x, y)$ with improved error term. Our analysis applies uniformly to the case where $x$ and $y$ are arbitrary close, and in particular allows for a meaningful conparison with the work of \cite{TH} in case $y=2x^2$, while keeping the computational complexity low. The method relies on the use of smooth cutoff functions, which provide greater flexibility in contour shifting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18395 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The second moment of Ramanujan sums Ziwei, Hong Zhiyong, Zheng Number Theory In this paper, we study $C(x, y)$, the second moment of Ramanujan sums. Assuming the Riemann Hypothesis(RH), we establish an asymptotic formula for $C(x, y)$ with improved error term. Our analysis applies uniformly to the case where $x$ and $y$ are arbitrary close, and in particular allows for a meaningful conparison with the work of \cite{TH} in case $y=2x^2$, while keeping the computational complexity low. The method relies on the use of smooth cutoff functions, which provide greater flexibility in contour shifting. |
| title | The second moment of Ramanujan sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.18395 |