On the Error Term for the Fourth Moment of the Riemann Zeta‐Function

Let E 2 ( T ) denote the error term in the asymptotic formula for ∫ T 0 ∣ζ(½+ it )∣ 4 d t . It is proved that there exist constants A >0, B >1 such that for T ⩾ T 0 >0 every interval [ T , BT ] contains points T 1 , T 2 for which∫ 0 T 1E 2 ( t ) d t > A T 1 3 / 2,∫ 0 T 2E 2 ( t ) d t < ‐ A T 2 3 / 2,and that ∫ T 0 ∣ E 2 ( t )∣ a d t ≫ T 1+( a /2) for any fixed a ⩾1. These results complement earlier results of Motohashi and Ivić that ∫ T 0 E 2 ( t )d t ≪ T 3/2 and that ∫ T 0 E 2 2 ( t )d t ≪ T 2 log C T for some C >0. Omega‐results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of ∣ζ(½+ it )∣ 4 .