Zeroes of the Zeta Function - young.physics.ucsc.edu · Zeroes of the Zeta Function This handout...

Zeroes of the Zeta Function This handout illustrates Mathematica's wide knowledge of mathematical functions. The zeta function Ζ(n) for positive integer n is defined by Ζ HnL = 1 + 1 2 n + 1 3 n + 1 4 n + ... It is well known that this blows up for n = 1 because the series is logarithmically divergent. Values of Ζ{n) are known exactly for n an even positive integer, e.g. Zeta@2D Π 2 6 The definition given above also makes sense if one replaces n by complex z provided that Re[z] > 1 but blows up if Re[z] < 1. However, there are other definitions of Ζ(z), beyond the scope of the course, which agree with the above one where the latter is valid but which can be used anywhere in the complex plane (except for z=1 exactly where the function has a pole). The properties of Ζ(z) in the complex plane, thus defined, have aroused a lot of interest, especially the location of the zeroes. There are "trivial" zeroes for z a negative even integer. This is illustrated in the plot below: Plot@Zeta@xD, 8x, -10.5, -1<, PlotRange ﬁ 88-10.5, 0<, 8-0.02, 0.01<<, AxesLabel ﬁ 8"x", "ΖHxL"<D -10 -8 -6 -4 -2 x -0.020 -0.015 -0.010 -0.005 0.005 0.010 Ζ HxL It is generally believed, but not proved, that all the other zeroes have Re[z] = 1/2. This is the famous Riemann hypothesis. Using Mathematica's ability to compute and plot functions in the complex plane, we can see that there are indeed zeroes along this line.

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Transcript of Zeroes of the Zeta Function - young.physics.ucsc.edu · Zeroes of the Zeta Function This handout...

Zeroes of the Zeta Function

This handout illustrates Mathematica's wide knowledge of mathematical functions.

The zeta function Ζ(n) for positive integer n is defined by

Ζ HnL = 1 +1

2n+

3n+

4n+ ...

It is well known that this blows up for n = 1 because the series is logarithmically divergent. Values of Ζ{n) are known exactly forn an even positive integer, e.g.

Zeta@2DΠ2

The definition given above also makes sense if one replaces n by complex z provided that Re[z] > 1 but blows up if Re[z] < 1.However, there are other definitions of Ζ(z), beyond the scope of the course, which agree with the above one where the latter isvalid but which can be used anywhere in the complex plane (except for z=1 exactly where the function has a pole).

The properties of Ζ(z) in the complex plane, thus defined, have aroused a lot of interest, especially the location of the zeroes.There are "trivial" zeroes for z a negative even integer. This is illustrated in the plot below:

Plot@Zeta@xD, 8x, -10.5, -1<, PlotRange ® 88-10.5, 0<, 8-0.02, 0.01<<, AxesLabel ® 8"x", "ΖHxL"<D

-10 -8 -6 -4 -2x

-0.020

-0.015

-0.010

-0.005

0.005

0.010ΖHxL

It is generally believed, but not proved, that all the other zeroes have Re[z] = 1/2. This is the famous Riemann hypothesis. UsingMathematica's ability to compute and plot functions in the complex plane, we can see that there are indeed zeroes along this line.

Page 2: Zeroes of the Zeta Function - young.physics.ucsc.edu · Zeroes of the Zeta Function This handout illustrates Mathematica's wide knowledge of mathematical functions. The zeta function

PlotBAbsBZetaB12

+ ä yFF, 8y, 0, 40<, AxesLabel ® 8"y", "ÈΖH1�2 + iyLÈ"<F

10 20 30 40y

0.5

1.0

1.5

2.0

2.5

3.0ÈΖH1�2 + iyLÈ

We can locate the roots accurately using the FindRoot command, taking the initial estimates (2 are needed because there is noanalytic expression for the derivative). As an example, for the first root:

y �. FindRoot@Abs@ Zeta@1�2 + I yDD, 8y, 14, 15<D14.1347

It can be shown that all zeroes but the trivial ones must lie the in the region 0 < Re[z] < 1. If you can find a zero in this regionwhich does not have Re[z] = 1/2 you will be famous!

Below is a 3-D plot of 1/| Ζ(x +I y)| where the zeroes of Ζ show up as peaks. The range covered is -2 < x < 2, 0 < y < 30. One seesthe expected peaks at x = 1/2 (three of them for this range of y) and the peak corresponding to the trivial zero at x = -2, y = 0.

Plot3DB 1

Abs@Zeta@x + ä yDD, 8x, -1.999, 2<, 8y, 0, 30<, PlotPoints ® 50,

PlotRange ® 80, 20<, ViewPoint ® 8-30, 10, 7<, AxesLabel ® 8"x", "y", "1�ÈΖHx+iyLÈ"<F

-2-1

0102030 y

1�ÈΖHx+iyLÈ

2 zeta.nb