Mixed character sums modulo prime powers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908934553468928 |
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| author | Cochrane, Todd Granville, Andrew |
| author_facet | Cochrane, Todd Granville, Andrew |
| contents | We obtain explicit estimates for the mixed character sum $S= S(χ,g,f,p^m) = \sum_{x=1}^{p^m} χ(g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $χ$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over $\mathbb Q$. Let $f=f_+/f_-$, $g=g_+/g_-$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_-g_+g_-$, and $Δ= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $Δ=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has
$|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$ if $\text{deg}_p(f) \ge 1$, and $|S| \le 3^{4/3}\, p^{m(1-\frac 1Δ)}$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mixed character sums modulo prime powers Cochrane, Todd Granville, Andrew Number Theory 11L07, 11L03 We obtain explicit estimates for the mixed character sum $S= S(χ,g,f,p^m) = \sum_{x=1}^{p^m} χ(g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $χ$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over $\mathbb Q$. Let $f=f_+/f_-$, $g=g_+/g_-$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_-g_+g_-$, and $Δ= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $Δ=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$ if $\text{deg}_p(f) \ge 1$, and $|S| \le 3^{4/3}\, p^{m(1-\frac 1Δ)}$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums. |
| title | Mixed character sums modulo prime powers |
| topic | Number Theory 11L07, 11L03 |
| url | https://arxiv.org/abs/2604.02614 |