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

Thi Altenschmidt

Published 2023-04-15Version 1

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 $$\int_0^T \Bigl|\zeta\left(\frac{1}{2}+it\right)\Bigr|^6dt \ll_{\varepsilon} T^{1+\varepsilon}, \quad T \to +\infty, \forall \varepsilon > 0.$$