A limit theorem for the arguments of zeta-functions of certain cusp forms (Q996740)

Let \(F(z)\) be a cusp form of weight \(\kappa\) for the full modular group \[ \text{SL}(2,\mathbb Z):=\Big\{\left(\begin{matrix} a & b\\ c & d \end{matrix}\right) : a,b,c,d \in\mathbb Z,ad-bc=1\Big\}. \] Then \(F(z)\) is holomorphic in \(\{z \in\mathbb C :\text{Im}(z) >0\}\), \(\lim_{\text{Im}(z) \to\infty} F(z)=0\) and \[ F\Big({{az+b}\over {cz+d}}\Big)=(cz+d)^{\kappa} F(z) \] for all \(\left (\begin{matrix} a & b\\ c & d\end{matrix}\right)\in \text{SL}(2,\mathbb Z).\) The authors also suppose that \(F(z)\) is a normalized eigenform for all Hecke operators. Then \(F(z)\) has the Fourier series expansion \[ F(z)=\sum_{n=1}^{\infty}c(m)e^{2\pi imz},\quad c(1)=1 \] with a multiplicative arithmetical function \(c(m)\). Therefore, \[ \varphi (s,F):=\sum_{m=1}^{\infty}{c(m)\over{m^s}} \quad (s \in \mathbb C) \] has an Euler product. Moreover, one can analytically continue \(\varphi (s,F)\) to an entire function and prove the functional equation \[ (2\pi)^{-1}\Gamma(s)\varphi(s,F)=(-1)^{\kappa/2}(2\pi)^{s-\kappa}\Gamma(\kappa -s)\varphi (\kappa -s,F). \] The critical strip for the nontrivial zeros of \(\varphi(s,F)\) is \(\{s\in {\mathbb C}:(\kappa-1)/2<\text{Re}(s)<(\kappa+1)/2\}\) and has for an analogue of the Riemann hypothesis the critical Line \(\{s\in\mathbb C : \text{Re}(s)=\kappa/2\}\). \textit{R. Ivanauskaitė} and \textit{A. Laurinčikas} [Chebyshevskiĭ\ Sb. 5, No. 4(12), 144--154 (2004; Zbl 1144.11040)] obtained a limit theorem for \(|\varphi (\sigma_T +it,F)|^{k_T}\) with \(\sigma_T \to{{\kappa}\over 2}, k_T \to 0\) as \(T\to\infty\). Using the moment method, the authors prove a limit theorem for \(|\arg\varphi({{\kappa}\over 2}+ it,F)|\), where this function is defined by a suitable continuous variation. Theorem. The distribution function \[ {1\over T}\text{meas}\{t\in[0,T]:{{|\arg\varphi({{\kappa}\over 2}+ it,F)|}\over \sqrt {2^{-1}\log \log T}}<x\} \] converges pointwise to \(\Phi_I(x)\) as \(T\to\infty\), where \[ \Phi_I(x)=\begin{cases} {2\over \sqrt {2\pi}}\int_{-\infty}^xe^{-{u^2}\over 2}\,du \,-1&\text{for }x>0,\\ 0 & \text{for }x\leq 0.\end{cases} \]