arXiv 1701.010422017-01-06math.NT

大奇阶character和与Pólya-Vinogradov不等式的改进

Large odd order character sums and improvements of the Pólya-Vinogradov inequality

Youness Lamzouri, Alexander P. Mangerel

备注：: 34 pages, fixed some typos

摘要：

对于本原狄利克雷特征标$χ$模$q$，我们定义$M(χ)=\max_{t } |\sum_{n \leq t} χ(n)|$。在本文中，我们研究了固定奇数阶$g\geq 3$的特征标的这个量。我们的主要结果在这种情况下提供了经典的Pólya-Vinogradov不等式的进一步改进。更具体地说，我们表明，对于任何这样的字符$χ$，我们有$$M(χ)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-δ_g}(\log\log q)^{-1/4+\varepsilon},$$，其中$δ_g := 1-\frac{g}π\sin(π/g)$。这改进了Granville和Soundararajan和Goldmakher的作品。此外，假设广义Riemann假设（GRH），我们证明了$$ M(χ) \ll \sqrt{q} \left(\log_2 q\right)^{1-δ_g} \left(\log_3 q\right)^{-\frac{1}{4}}\left(\log_4 q\right)^{O(1)}, $$，其中$\log_j$是第$j$次重对数。我们还无条件地表明，这个界是最好的可能（高达$\log_4 q$的权力）。证明上界的关键因素之一是一个新的Halász型不等式的对数平均值的完全乘法函数，这可能是独立的利益。

英文摘要：

For a primitive Dirichlet character $χ$ modulo $q$, we define $M(χ)=\max_{t } |\sum_{n \leq t} χ(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character $χ$ we have $$M(χ)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-δ_g}(\log\log q)^{-1/4+\varepsilon},$$ where $δ_g := 1-\frac{g}π\sin(π/g)$. This improves upon the works of Granville and Soundararajan and of Goldmakher. Furthermore, assuming the Generalized Riemann hypothesis (GRH) we prove that $$ M(χ) \ll \sqrt{q} \left(\log_2 q\right)^{1-δ_g} \left(\log_3 q\right)^{-\frac{1}{4}}\left(\log_4 q\right)^{O(1)}, $$ where $\log_j$ is the $j$-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of $\log_4 q$). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

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