Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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19. Turning the Golden Key
19. Turning the Golden Key
Pages 296-311
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From page 296... ...
My hope is that you will end up with at least an outline of the main logical steps Riemann followed. I can't do even that much, though, without a very small amount of calculus, the essential points of which I have already laid out in Chapter 7.vi-vii.
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From page 297... ...
Step functions are a little hard to get used to at first, but from a mathematical point of view they are perfectly sound. The domain here is all nonnegative numbers.
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From page 298... ...
III. Now I am going to introduce another function, also a step function, just a little bit odder than Fez (x)
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From page 299... ...
The square root of 100 is 10; the cube root is 4.641588...; the fourth root is 3.162277...; the fifth root is 2.511886...; the sixth root is 2.154434...; the seventh root is 1.930697...; the eighth root is 1.778279...; the ninth root is 1.668100..., and the tenth root is 1.584893.... I could, of course, go on working out the eleventh, twelfth, thirteenth roots, and so on for as far as you please.
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From page 300... ...
You can see that the Ifunction jumps suddenly from one value to another, holds the new value for a while, then makes another jump. What are these jumps?
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From page 301... ...
. At the actual point where a jump occurs, the function value is halfway up the jump.
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From page 302... ...
and an internet search will turn up numerous references. Rather like the Fez and ~ functions themselves, instead of gliding smoothly from one point to the next, I am going to vault over to the following fact: When Mobius inversion is applied to Expression 19-1, the result is as shown in Expression 19-2.
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From page 303... ...
This is because the Ifunction, like the Fez function, is zero when x is less than 2 (check the graph) , and if you keep taking roots of a number, the answers eventually drop below 2 and stay there.
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From page 304... ...
1 (1 1 ) Now recall Sir Isaac Newton's infinite series for log ( 1 - x)
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From page 305... ...
From a fairly neat little infinite product, I have now got myself an infinite sum of infinite sums. The situation might seem hopeless.
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From page 306... ...
The image it conjures up ~ the one in Figure 19-4. IEs the 7~ncdon' with ~ strip filled in.
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From page 307... ...
See how it all keys in to that infinite sum of infinite sums in Expression 19-3? Of course, the area under the ~ function is infinite.
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From page 308... ...
The sum of the squished strips, considering just the prime 2, iS shown in Expression 19-4.
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From page 309... ...
00 00 00 Ilxx-s-ldx+12 XX-s-ldX+T 1 Xx-s-ldx 3 32 r 33 oo + 1 1 x x-s-ldx + 1 5 x x-s-ldx + 3J4 35 Expression 19-5 There's another for 5, another for 7, and so on for all the primes. An infinite sum of infinite sums of integrals!
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From page 310... ...
Which is exactly what Riemann had set out to do, because then all the properties of the Fez function will be found encoded somehow in the properties of the ~ function. The Fez function belongs to number theory; the ~ function belongs to analysis and calculus; and we have just thrown a pontoon bridge across the gap between the two, between counting and measuring.
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From page 311... ...
x-s-~dx, for s= 1.2. Its numerical value is actually 1.434385276163....
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