The sixth power moment of partial Riemann zeta function and Kloosterman sums (Q6619680)

For \(T>0\), the following mean value for the partial sum \(Z(t,M)\) of the Riemann zeta function on the critical line is discussed.\N\N\[\N\tag{1} \mathcal{M}(T,M)=\int_T^{2T}|Z(t,M)|^6\,{\mathrm d}t,\N\]\Nwhere \(Z(t,M)=\sum_{M<m\le M'}m^{-1/2-it}\), \(M< M'\le 2M\) and \(T^{1/3}<M<T^{1/2}\). The bound \(\mathcal{M}(T,M)\ll T^{1+\varepsilon}, \forall \varepsilon>0\) is expected, which would be a step towards supporting the \(6\)th power moment of the Riemann zeta function on the critical line.\N\NThe goal of this project is a somewhat weaker bound\N\[\N\tag{4} \mathcal{M}(T,M)\ll H^{1/2}T^{1+\varepsilon}, \quad H=M^{3}T^{-1},\N\]\Nwhich implies an improvement in the zero-density estimation. Some plausible conjectures on estimates on Kloosterman sums are stated to deal with the hardest case \(M\approx T^{2/5}, H=T^{1/5}\).