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

Snapshot 3:

for

, showing that

,

, and

. It is surprising that

is the last

for which

until

.

In number theory, we define

(capital omega) to be the number of prime factors of

, counting multiplicity. Therefore,

,

,

,

, and

. The Liouville lambda function

is defined to be

. So,

,

,

,

, and

.

The function

is defined to be the sum of

for

. Whenever

has an even number of prime factors (counting multiplicity), that

contributes

to the sum. Whenever

has an odd number of prime factors, that

contributes

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

:

,

where

is the

complex zero of the Riemann zeta function.

The first three complex zeros of the zeta function are approximately

,

, and

. 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 [1]).

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

and

. 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

. As

increases, this expression becomes more and more negative. This means that it must be the sum involving zeta zeros that causes the graph of

to fluctuate.

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

for all

. However, in 1958, this conjecture was proven to be false. We now know that the smallest counterexample

for which

is

. In fact,

.

All known zeta zeros have multiplicity 1, which is assumed in the formula. At those values of

where

jumps from

to

, the formula converges to the midpoint,

, as

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 [3]. The key is to apply Perron's formula to the following identity, which holds for

:

.

This identity is found in [2], or it may be derived (in

*Mathematica* Version 7 or higher) from

DirichletTransform[LiouvilleLambda[n], n, s]. [2] G. H. Hardy and E. M. Wright,

*An Introduction to the Theory of Numbers*, 4th ed., Oxford: Oxford University Press, 1965 p. 255.

[3] H. L. Montgomery and R. C. Vaughan,

*Multiplicative Number Theory: I. Classical Theory*, Cambridge: Cambridge University Press, 2007 p. 397.