Let $\chi$ be a primitive character modulo a prime $q$, and let $\delta >0$. It has previously been observed that if $\chi$ has large order $d⩾d_0(\delta )$ then $\chi (n)\ne 1$ for some $n⩽q^{\delta }$, 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 $\alpha$ the number of $n⩽x$ such that $\chi (n)=\alpha$ is $o_{d\to \infty }(x)$ whenever $x>q^{\delta }$. Consequently, when $\chi$ has sufficiently large order the sequence ${(\chi (n))}_{n⩽q^{\delta }}$ cannot cluster near $1$ for any $\delta >0$. Our proof relies on a second moment estimate for short sums of the characters ${\chi }^{\ell }$, averaged over $1⩽\ell ⩽d-1$, that is non-trivial whenever $d$ has no small prime factors. In particular, given any $\delta >0$ we show that for all but $o(d)$ powers $1⩽\ell ⩽d-1$, the partial sums of ${\chi }^{\ell }$ exhibit cancellation in intervals $n⩽q^{\delta }$ as long as $d⩾d_0(\delta )$ 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 $3⩽d⩽q-1$, the Pólya–Vinogradov inequality may be improved for ${\chi }^{\ell }$ on average over $1⩽\ell ⩽d-1$, extending work of Granville and Soundararajan.

中文翻译：

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大量高阶字符

让$\chi$是对素数取模的原始字符$q$， 然后让$\delta >0$。之前已经观察到，如果$\chi$有大订单$d⩾d_0（\delta ）$然后$\chi （n）\ne 1$对于一些$n⩽q^{\delta }$，类似于维诺格拉多夫关于二次非留数的猜想。我们对这个事实给出了一个新的、简单的证明。此外，我们表明，如果$d$那么对于任何 $d$统一根$\alpha$的数量$n⩽X$这样$\chi （n）=\alpha$是${哦}_{d\to \mathrm{无穷大}}（X）$每当$X>q^{\delta }$。因此，当$\chi$序列的阶数足够大${（\chi （n））}_{n⩽q^{\delta }}$不能聚集在附近$1$对于任何$\delta >0$。我们的证明依赖于对字符的短和的二阶矩估计${\chi }^{\ell }$, 平均超过$1⩽\ell ⩽d-1$，这在任何时候都是不平凡的$d$有不小的质因数。特别是，给定任何$\delta >0$我们向所有人展示了这一点，除了$哦（d）$权力$1⩽\ell ⩽d-1$，部分总和${\chi }^{\ell }$展览间歇期取消$n⩽q^{\delta }$只要$d⩾d_0（\delta ）$是素数，超出了伯吉斯定理。我们的论点将自命不凡的数论和加性组合数学的结果融合在一起。最后，我们证明，均匀地超过素数$3⩽d⩽q-1$，Pólya-Vinogradov 不等式可能会得到改善${\chi }^{\ell }$平均超过$1⩽\ell ⩽d-1$，扩展了 Granville 和 Soundararajan 的工作。