Snapshot 1: the graphs of the step function

and the formula using no zeta zeros

Snapshot 2: the graphs of

and the formula using 30 pairs of zeta zeros

Snapshot 3: in order to hit the small jump in

at

, at least 70 zeta zeros are needed

can be defined as

, where the sum is over the primes

, and

denotes the floor function. For example,

=

=

.

Note that 2520 is the least common multiple of the integers 1 through 10.

behaves as follows: when

reaches a prime power

, then

increases by

. After that,

remains constant until

reaches the next prime power. This means the graph of

is a step function that jumps up at powers of primes.

This Demonstration uses von Mangoldt's formula to calculate

:

,

where

is the

complex zero of the Riemann zeta function. See [1] for general information about the

function. See p. 104 of [2] for a proof of this formula.

is a constant that simplifies to

, which is about 1.837877;

is the dominant term in the formula. It can be proven that as

approaches

, the ratio

approaches 1. This fact is equivalent to the prime number theorem, which states that as

approaches

, the ratio

approaches 1, where

is the number of primes less than or equal to

.

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 [3]).

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.

Notice that the second sum has the same form as the first, except that the second sum extends over the so-called "trivial" zeros of the zeta function, namely,

. The second sum is small when

is large.

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

. Since

jumps up by

at powers of the prime

, this means that the zeta zeros contain information about which numbers are primes or powers of primes.

In

*Mathematica* notation,

can be written as

psi[x_] := Sum[Floor[Log[Prime[k], x]] Log[Prime[k]], {k, 1, PrimePi[x]}].In number theory, the von Mangoldt lambda function

is defined as follows: If

is a power of a prime, say

, where

, then

is defined as

. For other values of

,

is 0. Using the von Mangoldt

function, we can also write

as

.

The von Mangoldt

function was introduced in Version 7 of

*Mathematica*. Using this function,

can be written more simply as

psi[x_] := Sum[MangoldtLambda[n], {n, 1, x}].[2] H. Davenport,

*Multiplicative Number Theory*, 3rd ed., New York: Springer, 2000.