On some conjectures and results for the Riemann zeta-function and Hecke series (Q2759124)

The topic of this paper is the (modified) Mellin transform NEWLINE\[NEWLINE\mathcal Z_2(s)=\int_1^{\infty}|\zeta (1/2+ix)|^4x^{-s} dx,NEWLINE\]NEWLINE introduced some years ago by Y. Motohashi as an important tool for the study of the fourth moment of the zeta-function. The author continues in some aspects a recent paper by \textit{A. Ivić}, \textit{M. Jutila} and \textit{Y. Motohashi} [Acta Arith. 95, 305-342 (2000; Zbl 0960.11039)], where the main problem was the estimation of \(\mathcal Z_2(s)\). As an improvement of a previous result, the author proves the estimate \(\mathcal Z_2(\sigma +it) \ll t^{1-\sigma + \varepsilon}\) for \(0 < \sigma <1\), \(t\geq 1\) at points well separated from the poles of \(\mathcal Z_2(s)\). This is a remarkable result, for it implies the best known estimate for the error term in the asymptotic formula for the fourth moment of the zeta-function, and any essential improvement would imply new information on the zeta-function. For instance, the validity of the hypothetical ``Lindelöf'' estimate \(\mathcal Z_2(\sigma +it)\ll t^{\varepsilon }\) for all fixed \(\sigma > 1/2\) would entail the essentially best possible estimate for the eighth moment of the zeta-function, and in fact it would suffice to know this hypothesis just in the mean square sense. Such a result would follow from a plausible but probably very deep spectral theoretic hypothesis, which the author formulates as his Conjecture 3.