In 1859, Riemann introduced the Zeta function in order to better understand the distribution of prime numbers. His famous conjecture concerning the location of the zeros of Zeta motivated intensive research on computing the first zeros of Zeta, counting their number in a box, finding zero-free regions, etc. Around the 1930’s, the analytic methods became complicated and the problem to make certain asymptotic results totally explicit arose. Nowadays, it is emerging that certain questions can be solved by a combination of explicit results in conjunction with the computing power of machines. In this talk, I will review some of the classical explicit results in analytic number theory:
1) How small is the error term in the prime number theorem?
2) How large does an interval around x have to be so that it contains a prime in an arithmetic progression?
3) What is the size of the smallest prime in an arithmetic progression?
4) Is every natural number more than or equal to 455 the sum of at most 7 non-negative cubes?
5) Is every odd natural number the sum of at most 3 primes?