The Dirichlet \(L\)-function \(L(s,\chi)\), \(s=\sigma+it\), is defined by \[ L(s, \chi) =\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}=\prod_{p\, \mathrm{ prime}}\bigg(1 - \frac{\chi(p)}{p^s}\bigg)^{-1}, \quad \sigma>1. \] The Generalized Riemann Hypothesis (GRH) states that if \(\varrho=\beta+i\gamma\) is a zero of \(L(s, \chi)\) with \(\beta > 0\), then \(\beta =\frac{1}{2}\). It is well known that, for any principal character \(\chi_0 \mod q\), \[ L(s,\chi_0)=\zeta(s)\prod_{p |q}\bigg(1-\frac{1}{p^s}\bigg), \] and, in this case, the Riemann hypothesis (RH) is equivalent to GRH for \(L(s,\chi_0)\) (here, as usual, \(\zeta(s)\) denotes the Riemann zeta-function).
In this paper, it is shown the weak form of GRH for Dirichlet \(L\)-functions (denote as GRH[\(\frac{9}{10}\)]): the inequality \(\beta \leq \frac{9}{10}\) holds for all zeros \(\varrho=\beta+i\gamma\) for an arbitrary Dirichlet \(L\)-function \(L(s,\chi)\). This is done entirely in terms of properties of the zeros of the function \(\zeta(s)\).
More precisely, the result follows from these facts. Suppose, that RH holds, and every function \(\mathcal{B}\) belongs to the space of smooth functions \(f: {\mathbb{R}}^+ \to {\mathbb{C}}\) with compact support in \(\mathbb{R}^+\), and every rational number \(\xi:=\frac{m}{q}\) is such that \(0<m<q\) and \((m,q)=1\). Let, for any \(\varepsilon>0\), it holds the bound \[ \sum_{\varrho=\frac{1}{2}+i\gamma}\xi^{-i\gamma}{\mathcal{Z}}(\varrho){\mathcal{B}}\bigg(\frac{\gamma}{2\pi X}\bigg)+\frac{\mu(q)}{\phi(q)}\sum_{n=1}^{\infty}\Lambda(n){\mathcal{B}}\bigg(\frac{n}{X}\bigg)\ll_{\xi,{\mathcal{B}},\varepsilon}X^{\frac{9}{10}+\varepsilon}, \] where \({\mathcal{Z}}(\varrho):=\lim_{T\to \gamma}{\overline{e^{2 i \arg \zeta\big({1}/{2}+iT\big)}}}\) with non-odinate \(T>0\) of a zero of the zeta-function, and first sum on left side runs over all complex zeros \(\varrho=\frac{1}{2}+i\gamma\) of \(\zeta(s)\). Under such conditions the hypothesis GRH[\(\frac{9}{10}\)] is true.