Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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13. The Argument Ant and the Value Ant
13. The Argument Ant and the Value Ant
Pages 201-222
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From page 201... ...
Table 13-1 shows a sample of the squaring function for some random complex numbers.67 TABLE 13-1 The Squaring Function.
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From page 202... ...
z en -1 + 2.141593i 3.141593i 1 + 4.141593i 2 + 5.141593i 3 + 6.141593i -0.198766 + 0.30956i -1 -1.46869 - 2.28736i 3.07493 - 6.71885i 19.885 - 2.83447i Note that, just as before, when I choose the arguments to go up by addition as of course I do, in this case adding 1 + i each timethe function values go up by multiplication, in this case by 1.46869 + 2.28736i. If I had picked the arguments to go up by adding 1 each time, then of course the values would have multiplied by e.
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From page 203... ...
This is because, in the complex world, the exponential function sometimes gives the same value for different arguments. The cube of-1, for example, is, by the rule of signs,-1; so if you cube both sides of em = -1, you get em = -1; so the arguments Fez i and 3 Fez i both yield the same function value of -1, just as -2 and +2 both yield value 4 under the squaring function.
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From page 204... ...
while the function values go up by addition (of 0.346574 + 0.785398i each time)
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From page 205... ...
As another example, since e~i=-l, taking the square rim root of both sides gives i = e 2 . If you now raise both sides to the Liz power of i, remembering Power Rule 3 again, you get ii = e 2 .
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From page 206... ...
How then can complex functions be visualized? Let's take the simplest non-trivial complex function, the squaring function.
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From page 207... ...
The function values need another two-dimensional plane. So to get a graph, you need four dimensions of space to draw it in: two for the argument, two for the function value.
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From page 208... ...
. Well, the argument number is a point somewhere on the complex plane; and the function value is some other point.
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From page 209... ...
Bernhard Riemann, who seems to have had a very powerful visual imagination, conceived of the matter like this. Take the entire complex plane.
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From page 210... ...
In one of the frontmost appendages, which for convenience we may call a"hand," the argument ant hol(ls a small gadget rather like a beeper, or a mobile telephone, or one of those global positioning devices that can tell you
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From page 211... ...
The second (lisplay, labele(l"Argument," shows the point the complex number the argument ant is currently standing on. The third display, labeled "Function value," shows the value of the function at that argument.
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From page 212... ...
At the point where each curve leaves the diagram at left or right, I have written in the function value corresponding to that point. Trying to imagine what the zeta function does to the complex plane in the sense of Figure 13-3, which shows what the squaring function does to it is a rather demanding mental exercise.
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From page 213... ...
1.000... 0.999...FIGURE 13-6 The argument plane, showing points that zeta "sends to" the real and imaginary axes.
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From page 214... ...
As he steps on the number 1, a buzzer goes off in the gadget he is hol(ling, an(1 the "Function value" (lisplay shows a big bright re(1 flashing infinity sign, "oo." If he looks more closely at the (lisplay, the argument ant will notice a curious thing. At the right of the infinity sign, a small letter")
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From page 215... ...
If, on the other hand, he takes a sharp right turn northwards at 1 and traverses the top half of that oval shape around the zero point, he will find from the display that the function values are ascending the negative imaginary axis, from numbers like-l,OOO,OOOi, up through -l,OOOito -lOi, -5i, -2 i, then to -i. Shortly before he crosses the imaginary axis the display reads-0.5)
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From page 216... ...
The next maps into the positive imaginary axis; the next, into the positive real axis; the next, into the negative imaginary axis, .
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From page 217... ...
In fact the average spacing between the eight zeros shown here is 2.096673119.... For the five zeros shown in Figure 13-6, the average spacing was 4.7000841....
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From page 218... ...
If I extended Figure 13-6 down south of the real axis, the lines would be mirror images of what they are north of it. The only difference is that while the real numbers I have written in on Figure 13-6 are just the same south as they are north, the imaginary numbers have their signs flipped.
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From page 219... ...
I had him wandering over the argument plane, noting where the argument points are ~ 1 ~ 1 1 1 ~ sent to by the zeta function. I actually had him wandering along strange curves and loops, made up of points that are sent to (i.e., whose function values are equal to)
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From page 220... ...
Now suppose that the argument ant, instead of following those fancy loops and whorls in Figure 13-6 (which send the value ant on dull hikes up and down the real and imaginary axes) , takes a walk straight up the critical line, hea(ling (lue north from argument -.
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From page 221... ...
" means "all arguments whose zeta function values are on the real or imaginary axis." Note that the expression " (-I" is used here in the special function-theory sense of "inverse function." Don't confuse it with a-i as in Power Rule 8, which has the meaning 1/a, the arithmetic reciprocal of a. This is a different usage another case of overloading math symbols, like the use of Fez for both 3.14159...
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From page 222... ...
How do we know that the trillionth one, or the trillion trillionth, or the trillion trillion trillion trillion trillion trillionth lies on the critical line? We don't, not from drawing diagrams anyway.
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