Bounding the exponential sum on squares of some sifted sequences
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910119594295296 |
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| author | Malavika, E. Ramaré, Olivier |
| author_facet | Malavika, E. Ramaré, Olivier |
| contents | Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09448 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounding the exponential sum on squares of some sifted sequences Malavika, E. Ramaré, Olivier Number Theory Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences. |
| title | Bounding the exponential sum on squares of some sifted sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.09448 |