论文标题
大量的高级字符
Large Sums of High Order Characters
论文作者
论文摘要
令$χ$为原始角色modulo a Prime $ Q $，让$δ> 0 $。以前已经观察到，如果$χ$具有大订单$ d \ geq d_0（δ）$，则与Vinogradov对Quadratic非官方的猜想相比，对于某些$ n \ leq q^δ$，$χ（n）\ neq 1 $。我们给出了这个事实的新简单证明。此外，我们还表明，如果$ d $是平方英语，那么对于任何$ d $ th的unity $α$的根，$ n \ leq x $的数量是$χ（n）=α$是$ o_ o_ o_ {d \ to \ to \ to \ to \ infty}（x）$。因此，当$χ$具有足够大的序列时，序列$（χ（n））_ {n \ leq q^δ} $无法聚集到任何$δ> 0 $的$ 1 $。 我们的证明依赖于第二刻的估算值，用于$ 1 \ leq \ ell \ el \ leq d-1 $的平均$χ^\ ell $的短款项，这是每当$ d $没有小的质量因素时，这是无处不在的。特别是，在任何$Δ> 0 $的情况下，我们表明除$ o（d）$ powers $ 1 \ leq \ ell \ leq d-1 $外，$χ^\ ell $的部分总和在间隔$ n \ leq leq q^δ$显示出$ d \ d \ egeq d_0（δ）$超过burgess burgess burgess。我们的论点将自命不凡的数字理论和添加剂组合学的结果融合在一起。 最后，我们表明，在Prime $ 3 \ leq d \ leq Q-1 $上均匀地，Pólya-Vinogradov的不平等可能会以$χ^\ ell $的平均$ 1 \ leq \ ell \ ell \ ell \ leq d-1 $的平均而提高，从而扩展了Granville和Soundararajan的工作。
Let $χ$ be a primitive character modulo a prime $q$, and let $δ> 0$. It has previously been observed that if $χ$ has large order $d \geq d_0(δ)$ then $χ(n) \neq 1$ for some $n \leq q^δ$, in analogy with Vinogradov's conjecture on quadratic non-residues. We give a new and simple proof of this fact. We show, furthermore, that if $d$ is squarefree then for any $d$th root of unity $α$ the number of $n \leq x$ such that $χ(n) = α$ is $o_{d \to \infty}(x)$ whenever $x > q^δ$. Consequently, when $χ$ has sufficiently large order the sequence $(χ(n))_{n \leq q^δ}$ cannot cluster near $1$ for any $δ> 0$. Our proof relies on a second moment estimate for short sums of the characters $χ^\ell$, averaged over $1 \leq \ell \leq d-1$, that is non-trivial whenever $d$ has no small prime factors. In particular, given any $δ> 0$ we show that for all but $o(d)$ powers $1 \leq \ell \leq d-1$, the partial sums of $χ^\ell$ exhibit cancellation in intervals $n \leq q^δ$ as long as $d \geq d_0(δ)$ is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime $3 \leq d \leq q-1$, the Pólya-Vinogradov inequality may be improved for $χ^\ell$ on average over $1 \leq \ell \leq d-1$, extending work of Granville and Soundararajan.
