Sharp upper bound for the sixth moment of the Riemann zeta function on the critical line

Abstract

The main task of this work is to give an improvement for the upper bounds of the Laplace transform $$\int_0^{+\infty}\Bigl|\zeta\left(\frac{1}{2}+it\right)\Bigr|^{2\beta}e^{-\delta t}dt \ll_{\beta,\varepsilon} \frac{1}{\delta^{\frac{\beta-1}{2}+\varepsilon}}, \quad 0 < \delta < \frac{\pi}{2}, \delta \to 0^+, \forall \varepsilon > 0, \forall \beta \geqslant 3.$$ In particular, this implies the desired estimation for the upper bound of the sixth moment of the Riemann zeta function on the critical line $$\int_0^T \Bigl|\zeta\left(\frac{1}{2}+it\right)\Bigr|^6dt \ll_{\varepsilon} T^{1+\varepsilon}, \quad T \to +\infty, \forall \varepsilon > 0.$$