Fractional moments of the \(\zeta\)-function
The author considers \(2\lambda\)-th (\(\leq 4)\) fractional moments of \(\zeta\) (s) in the right half of the critical strip. He proves the following theorem. Let \(0<\lambda \leq 2\), \(T\geq 10^ 3\), \(\sigma \geq 1/2+(\log \log \log T)^ 2/\log \log T\). Then we have the formula \[ \int^{T}_{1}| \zeta (\sigma +it)|^{2\lambda} dt=C(\sigma,\lambda)T+O(\Delta (T^{1-(2\sigma -1)/\quad 2(3- 2\sigma)}+T^{1-(2\sigma -1)/(2-\sigma)}(1-\lambda (1-\sigma)))), \] \(C(\sigma,\lambda)=\sum^{\infty}_{n=1}\tau^ 2_{\lambda}(n) n^{- 2\sigma}\), \(\tau_{\lambda}(n)\) are the coefficients of the Dirichlet series for \(\zeta^{\lambda}(s)\) for Re s\(>1\), and \(\Delta =Exp\{C_ 0 (\log \log \log T)^ 2/\log \log T\}\), \(C_ 0>0\) being an absolute constant.
scientific article; zbMATH DE number 3950634