The zeta function, denoted

, is a function of the complex variable

, where

and

. It was first introduced and studied early in the eighteenth century by Leonhard Euler [1], who computed it for some positive and negative integer values of

[2]. Later work by Pafnuty Chebyshev extended the

function's domain to all

[2]. Bernhard Riemann [3] further extended

by analytic continuation to all complex values, including

, and today

is generally referred to as the Riemann zeta function [2–6].

This Demonstration can plot the first 24 trivial and critical zeros of

in five different ways. They are chosen from the "plot" popup menu as described in the paragraphs below. Not all of the 15 controls apply to every plot choice.

The dragging options and the blue "restore initial 3D view" setter control apply only to the 3D graphics that are seen in plots 3, 4 and 5. You can rotate any 3D plot by dragging on it. To translate (slide-move) the plot, Shift+drag on it. To zoom in or out use Alt+drag (in Windows) or Option+drag or Command+drag (in macOS). To zoom in, drag bottom to top; to zoom out, drag top to bottom. To cancel the effects of any dragging, click the blue setter, which returns the 3D plot to the original viewpoint.

Select plot option "1: trivial zeros 2D plot" to display trivial zeros on the 2D axes

versus

. The "

axis minimum" slider sets the minimum value of the

axis. It also varies the number (from 0 to 24) of trivial zeros plotted as pink points at even negative integer values on the gray

axis. The dashed brown line at

passes through the brown pole point at

, which is the only place in the complex domain where

diverges and is undefined.

When the "plot

regularization point" box is checked, the point

is plotted in magenta on the negative

side of the blue

curve. The point seems to imply a surprising and counterintuitive equality, because substituting

in the infinite series sum that defines

reveals the following:

,

which seems to imply that

[7]. However, setting the left side of the above equation equal to

is an abuse of notation. Mathematica's notation makes the difference explicit. Evaluating

Sum[n,{n,1,∞}] in Mathematica returns a "Sum: Sum does not converge." error, as you would expect. Evaluating

Sum[n,{n,1,∞},Regularization->"Dirichlet"] returns

, making it clear that the two sums are not the same. However, the expression

Zeta[-1]==Sum[n,{n,1,∞},Regularization->"Dirichlet"]==-1/12 evaluates to True, showing that the value of

is indeed equivalent to the

*regularized* infinite sum of the positive integers. The value of

is in the analytically continued part of the complex domain of

, but because

, this regularized sum involves

*only* real numbers.

The regularized sum is elsewhere referred to as "zeta function regularization" [7], but in Mathematica it is referenced with the option

Regularization->"Dirichlet", partly because the infinite series that defines

is one example of the more general Dirichlet series

, where

is complex and the complex sequence

[8], but also because the first three

Regularization options to Mathematica's

Sum and

Product functions are

Abel,

Borel and

Cesaro, someone wanted the fourth option to be a word beginning with

"D", and

"Riemann" and

"Zeta" did not qualify [9]. The regularized product of the infinite series of positive integers can also be computed using that same option.

Product[n,{n,1,Infinity}] returns the expected error message "Product: Product does not converge.", but

Product[n,{n,1,Infinity},Regularization->"Dirichlet"] returns

.

Select plot option "2: critical zeros 2D plot" to display the real and imaginary parts of

versus

as green and brown curves, respectively. The critical zeros are shown as red points lying on the black

axis, which represents the domain line

, known as the critical line. Notice the characteristic periodicities of the two curves.

Select plot option "3: 2D view of critical zeros 3D plot" to display

on the axes

,

and

as viewed looking down the

axis toward the origin. Because the

* *axis is orthogonal to the screen, all the red critical zero points appear as a single point at the plot's origin [10, 11]. To see the points lying on the black critical line, click and drag on the image to rotate it in 3D or choose plot option 4.

Select plot option "4: critical zeros 3D plot" to display critical zeros on the same 3D axes as the plot 3 choice, but from a different viewpoint that allows you to see the curve in three dimensions without needing to rotate the plot. You can still drag and rotate the 3D viewpoint of plot 4 to match that of plot 3 and vice versa.

Select plot option "5: trivial and critical zeros 4D plot on 3D axes" to display both trivial and critical zero points on a 4D plot of

. The three axes are

modulus (magnitude) of

,

, and

. The fourth dimension is

(phase angle of

) plotted as a cyclic color function on the 3D surface of

. The

surface appears totally opaque at opacity 1. The trivial (pink) and critical (red) points and lines on which they lie become more visible through the colored ζ surface as the opacity value is set closer to 0 and the surface becomes invisible at opacity 0. Setting the "number

of critical zero pairs to plot" slider to 0 will remove the title and critical zeros coordinates and all trivial and critical zero points from the plot but will retain the plot range of all three axes that was used at the previous setting of

. With the "speed up plot 5" box unchecked the plot takes longer to compute but the pole peak at

becomes much more prominent, as seen in Snapshot 4.

The famous Riemann hypothesis of 1859 [4] asserts that all non-trivial zeros of

lie on the critical line. Over 10 trillion (

) critical zeros of

have been shown to conform [12]. However, the hypothesis itself remains unproven and proving it could win the proof's author a $1,000,000 Millennium Prize from the Clay Mathematics Institute [12]. A short and well-written recent book explains that the Riemann hypothesis implies that the position of the primes in the sequence of natural numbers is encoded in the imaginary parts of the non-trivial zeros along the critical line. [13]

Snapshot 1: Plot option 2 shows the first 12 critical zeros of

as red points on both the positive and negative sides of the

axis. The

versus

curve is green and the

versus

curve is brown.

Snapshot 2: Plot option 3 shows a 2D view of the first 24 positive critical zeros of

plotted on 3D axes

,

and

. The

curve is the thin blue line.

Snapshot 3: Plot option 4 shows six positive and six negative critical zeros of

plotted on 3D axes

,

and

. The

curve is the medium blue line; the

critical line is the

axis plotted as thin and black on which the critical zeros appear as large red points.

Snapshot 4: Plot option 5 shows a 4D plot of a single trivial zero, a single positive, and a single negative critical zero of

plotted on 3D axes

,

, and

where the color contours plot

as a fourth dimension. The "

surface opacity" slider is set to 1. The critical line is plotted as thick and black and the

axis line as thick and gray. The critical zero points are plotted as large and red and the trivial zero points as large and pink. The legend for

is plotted as a column to the right of the plot. The "speed up plot 5" box is unchecked so the pole peak at

is clearly visible, though the plot takes longer to complete.

[13] B. Mazur and W. Stein,

*Prime Numbers and the Riemann Hypothesis*, Cambridge: Cambridge University Press, 2016.