The mean value of Hardy's function in short intervals
For real \(t\), Hardy's function is defined by \[ Z(t)=\chi\left(\tfrac{1}{2}+it\right)^{-1/2} \zeta\left(\tfrac{1}{2}+it\right), \] where \(\zeta(s)\) is the Riemann zeta-function and \(\chi(s)\) is the gamma factor determined by \(\zeta(s)=\chi(s)\zeta(1-s)\). We note that \(Z(t)\) is a real-valued function and that \(|Z(t)|=|\zeta(\frac12+it)|\) holds for any real \(t\). In this paper, the author studies \[ I(T,H,U)=\int_T^{T+H}\left(\int_0^U Z(t+u)\,du\right)^2 dt, \] where \(0<U\leq T^{1/2}\leq H\leq T\). By three different approaches the author gives several formulas of the type \(I(T,H,U)=M(T,H,U)+E(T,H,U)\). Here \(M(T,H,U)\) is written explicitly and \(E(T,H,U)\) is regarded as an error term under certain condition among \(T\), \(H\) and \(U\).
scientific article; zbMATH DE number 6517638
scientific article