论文标题
耐火木 - 河畔型型式型式式Riemann假设的标准
Hardy-Littlewood-Riesz type equivalent criteria for the Generalized Riemann hypothesis
论文作者
论文摘要
在本文中，我们证明了dirichlet $ l $ function $ l（s，χ）$的广义Riemann假设等同于以下限制：让$ k \ geq 1 $和$ \ ell $为正实数。对于任何$ε> 0 $，我们有\ begin {align*} \ sum_ {n = 1}^{\ infty} \ frac {χ（n）μ（n）} {n^{k}}}}}}} \ exp \ exp \ left（ - o_ {ε，k，\ el} \ big（x^{ - \ frac {k} {\ ell} + \ frac {1} {1} {2 \ ell} +ε} \ bigg），\ \ quad \ quad \ quad \ quad \ quadrm {as}}}原始的dirichlet字符modulo $ q $，$μ（n）$表示möbius函数。这范围概括了Riesz和Hardy-Littlewood给出的以前的边界。
In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,χ)$ is equivalent to the following bound: Let $k \geq 1$ and $\ell$ be positive real numbers. For any $ε>0$, we have \begin{align*} \sum_{n=1}^{\infty} \frac{χ(n) μ(n)}{n^{k}} \exp \left(- \frac{ x}{n^{\ell}}\right) = O_{ε,k,\ell} \bigg(x^{-\frac{k}{\ell}+\frac{1}{2 \ell} + ε}\bigg), \quad \mathrm{as}\,\, x \rightarrow \infty, \end{align*} where $χ$ is a primitive Dirichlet character modulo $q$, and $μ(n)$ denotes the Möbius function. This bound generalizes the previous bounds given by Riesz, and Hardy-Littlewood.
