  Prime numbers seem to distribute among the natural numbers with no other law
than that of chance, however its global distribution presents a quite
remarkable smoothness. Such interplay between randomness and regularity has
motivated sci- entists of all ages to search for local and global patterns in
this distribution that eventually could shed light into the ultimate nature of
primes. In this work we show that a generalization of the well known
first-digit Benford's law, which addresses the rate of appearance of a given
leading digit d in data sets, describes with astonishing precision the
statistical distribution of leading digits in the prime numbers sequence.
Moreover, a reciprocal version of this pattern also takes place in the sequence
of the nontrivial Riemann zeta zeros. We prove that the prime number theorem
is, in the last analysis, the responsible of these patterns. Some new relations
concerning the prime numbers distribution are also deduced, including a new
approximation to the counting function pi(n). Furthermore, some relations
concerning the statistical conformance to this generalized Benford's law are
derived. Some applications are finally discussed.
