Wolfram Demonstrations Project

The Euler phi (

) function (also called the totient function) is important in number theory;

is the number of positive integers less than or equal to

that have no factor in common with

. For example,

and

The sum of

for positive integers

is a function of

that is usually denoted by

is an increasing, but irregular, step function.

This Demonstration illustrates the remarkable fact that we can approximate the jumps of this step function by using a sum that involves zeros of the Riemann zeta function

Contributed by: Robert Baillie

Slider Zoom
Automatic Animation

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: for larger

, we need more and more zeta zeros in order to closely match the step function

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.

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,