Snapshot 1: the graphs of the step function and the formula
using no zeta zeros Snapshot 2: the graphs of and the formula
using 40 pairs of zeta zeros The Moebius mu function is defined
to be 1 if ; it is is the product of distinct primes and 0
otherwise (i.e., if is divisible by the square of a prime). The
Mertens function is defined to be the sum of for positive
integers The first three complex zeros of the zeta function are
approximately . The zeros occur in conjugate pairs, so if is a
zero, then so is . The important Riemann hypothesis is the
unproven conjecture that all these complex zeros have real part
1/2. So far, it has been verified that the first complex zeros
do, indeed, have real part 1/2 (see ). In 1897, Mertens
conjectured that for is always between . If the Mertens
conjecture were true, then the Riemann hypothesis would be also
have to be true. But in 1985, Odlyzko and te Riele  proved that
the Mertens conjecture is false. This does not disprove the
Riemann hypothesis, which remains one of the