In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The first explicit results in this direction were proved by Turán and Knapowski in the 1950s, answering a problem of Littlewood from the year 1937. They used the power sum method of Turán. Our present approach does not use Turán’s method and gives sharper results.