On monotonicity of certain weighted summatory functions associated with \(L\)-functions (Q2910126)

Let \(\{c(n)\}_{n \in \mathbb N}\) be a sequence of real numbers, and let \(g: (0,\infty) \to \mathbb R\) be a real-valued locally integrable function. The weighed summatory function \(h(x)\) is given by NEWLINE\[NEWLINE h(x)=\sum_{n=1}^{\infty}c(n)g\bigg(\frac{n}{x}\bigg). NEWLINE\]NEWLINE In the paper, the author investigates the monotonicity of family of weighted summatory functions \(h_{f,w}^{<k>}: (0,\infty) \to \mathbb R\) associated with general \(L\)-functions \(L(f,s)\) in the sense of [\textit{H. Iwaniec} and \textit{E. Kowalski}, Analytic number theory. Providence, RI: AMS (2004; Zbl 1059.11001)] equipped with parameters \(0<\omega<1/2\) and \(k \in \mathbb N\). It is shown that, for arbitrary fixed \(k \geq 2\) and large \(x>0\), the monotonicity of \(\int_{1}^{x}h_{f,\omega}^{<k>}(t)\, dt\) is equivalent to the Grand Riemann hypothesis of \(L(f,s)\).