Hardy–Littlewood–Riesz type equivalent criteria for the Generalized Riemann hypothesis
نویسندگان
چکیده
In the present paper, we prove that generalized Riemann hypothesis for Dirichlet L-function $$L(s,\chi )$$ is equivalent to following bound: Let $$k \ge 1$$ and $$\ell $$ be positive real numbers. For any $$\epsilon >0$$ , have $$\begin{aligned} \sum _{n=1}^{\infty } \frac{\chi (n) \mu (n)}{n^{k}} \exp \left( - \frac{ x}{n^{\ell }}\right) = O_{\epsilon ,k,\ell \bigg (x^{-\frac{k}{\ell }+\frac{1}{2 \ell + \epsilon }\bigg ), \quad \textrm{as}\,\, x \rightarrow \infty \end{aligned}$$ where $$\chi a primitive character modulo q, $$\mu (n)$$ denotes Möbius function. This bound generalizes previous bounds given by Riesz, Hardy–Littlewood.
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ژورنال
عنوان ژورنال: Monatshefte für Mathematik
سال: 2023
ISSN: ['0026-9255', '1436-5081']
DOI: https://doi.org/10.1007/s00605-023-01857-8