Hardy-Littlewood-Riesz type equivalent criteria for the Generalized Riemann hypothesis

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,\chi )$ is equivalent to the following bound: Let $k\ge 1$ and $\mathrm{\ell }$ be positive real numbers. For any $\epsilon >0$, 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 $\mathrm{a}\mathrm{s}$,, x \rightarrow \infty, \end{align*} where $\chi$ is a primitive Dirichlet character modulo $q$, and $\mu (n)$ denotes the Möbius function. This bound generalizes the previous bounds given by Riesz, and Hardy-Littlewood.