On Large Oscillations of the Remainder of the Prime Number Theorems

Under the assumption of the appropriate Ricmann hypothesis it is shown that max _tx min _l1 t ^-1/2 ((x, q, 1) – (x, q, l)) > ( – ) log₃ x and min _tx max _l1 t ^-1/2 ((x, q, 1) – (x, q, l)) < –( – ) log₃ x for x > x ₀(q, ). The proof is quite elementary, and x ₀ can be estimated effectively. As a by-product a formula for the k-th power moment of certain normed error terms is obtained.