Snapshot 1: the graphs of the step function
and the formula using no zeta zeros
Snapshot 2: the graphs of
and the formula using 50 pairs of zeta zeros
, showing that
. It is surprising that
is the last
In number theory, we define
(capital omega) to be the number of prime factors of
, counting multiplicity. Therefore,
. The Liouville lambda function
is defined to be
is defined to be the sum of
has an even number of prime factors (counting multiplicity), that
to the sum. Whenever
has an odd number of prime factors, that
to the sum. This means that
is the number of integers in the range
that have an even number of prime factors, minus the number that have an odd number of prime factors.
This Demonstration uses the following formula to calculate
complex zero of the Riemann zeta function.
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 ).
If you use the slider to choose, say, one pair of zeta zeros, then the first sum in the above formula, in effect, combines two terms corresponding to the first conjugate pair of zeta zeros
. So, when these terms are added, their imaginary parts cancel while their real parts add. The
applied to the first sum is simply an efficient way to combine the two terms for each pair of zeta zeros.
The dominant part of the formula is
increases, this expression becomes more and more negative. This means that it must be the sum involving zeta zeros that causes the graph of
For example, at
. But when we include 100 zeta zeros, the above formula gives the value
, which is much closer to the exact value,
. We could get even closer to this exact value by taking larger values of
in the above formula.
You can see from the graphs that for small
is less than or equal to zero. In 1919, the mathematician George Pólya conjectured that
. However, in 1958, this conjecture was proven to be false. We now know that the smallest counterexample
. In fact,
All known zeta zeros have multiplicity 1, which is assumed in the formula. At those values of
, the formula converges to the midpoint,
approaches infinity. You can see this in the graphs.
As we include more zeta zeros in the first sum, we more closely replicate the jumps in the step function
. This means that the zeros of the Riemann zeta function somehow "know" how many prime factors each of the integers has.
To prove formulas like the one above, see . The key is to apply Perron's formula to the following identity, which holds for
This identity is found in , or it may be derived (in Mathematica
Version 7 or higher) from
DirichletTransform[LiouvilleLambda[n], n, s].
 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers
, 4th ed., Oxford: Oxford University Press, 1965 p. 255.
 H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory: I. Classical Theory
, Cambridge: Cambridge University Press, 2007 p. 397.