Distribution of differences of characters evaluated at consecutive polynomial values
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911404750012416 |
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| author | Bag, Nilanjan Mazumder, Dwaipayan |
| author_facet | Bag, Nilanjan Mazumder, Dwaipayan |
| contents | In this paper, we study the distribution of difference of multiplicative and additive characters modulo $p$ at consecutive polynomial values. More precisely, for an interval $I$ over finite field and $0<m<1$, we investigate the following sums \begin{align*}
\sum_{n\in I}|ψ(F(n))-ψ(F(n+1))|^{2m} \quad \text{and} \quad \sum_{n\in I}|χ(F(n))-χ(F(n+1))|^{2m}, \end{align*} where $ψ$ is a non-trivial additive character and $χ$ is a non-trivial multiplicative character modulo $p$, under suitable conditions on $χ$ and $F$. As a consequence, we derive a formula for the first moment by specializing to $m=1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of differences of characters evaluated at consecutive polynomial values Bag, Nilanjan Mazumder, Dwaipayan Number Theory 11L07, 11N37, 05A10 In this paper, we study the distribution of difference of multiplicative and additive characters modulo $p$ at consecutive polynomial values. More precisely, for an interval $I$ over finite field and $0<m<1$, we investigate the following sums \begin{align*} \sum_{n\in I}|ψ(F(n))-ψ(F(n+1))|^{2m} \quad \text{and} \quad \sum_{n\in I}|χ(F(n))-χ(F(n+1))|^{2m}, \end{align*} where $ψ$ is a non-trivial additive character and $χ$ is a non-trivial multiplicative character modulo $p$, under suitable conditions on $χ$ and $F$. As a consequence, we derive a formula for the first moment by specializing to $m=1/2$. |
| title | Distribution of differences of characters evaluated at consecutive polynomial values |
| topic | Number Theory 11L07, 11N37, 05A10 |
| url | https://arxiv.org/abs/2505.11859 |