Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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9. Domain Stretching
9. Domain Stretching
Pages 137-150
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From page 137... ...
Let me state it again, just as a refresher. The Riemann Hypothesis All non-trivial zeros of the zeta function have real part one-half.
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From page 138... ...
= H(l-p-s) -l p where the terms of the infinite product run through all the primes.
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From page 139... ...
= 1, since zero squared, zero cubed, and so on are all zero, and only the initial 1 is left standing. If x is 1, however, S(1)
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From page 140... ...
. Can it be that behind that infinite sum is the perfectly simple function 1/~1 - x)
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From page 141... ...
It all checks out. For all the arguments - 2 ~ - 3 ~ 0' 3 ~ 2, for which we know a function value, the value is the same for the infinite series S(x)
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From page 142... ...
, the function value Is very large but negative as if, when you cross the line s = 1 heading west, the value suddenly flips from infinity to minus infinity.
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From page 143... ...
The zeta function is zero at every negative even number, and the successive peaks and troughs now (Figures 9-5 to 9-10) get rapi(lly more and more dramatic as you head west.
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From page 145... ...
Mathematicians can prove, in fact though I'm not going to prove it here that this new infinite series converges whenever s is greater than zero. This is a big improvement on Expression 9-1, which converges only for s greater than 1.
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From page 146... ...
But hold on there a minute. How can I juggle these two infinite series at the argument s= -, where one of the series converges and one (loesn't?
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From page 147... ...
= 1 x 2 x 3 x 4, and so on. In advanced math, though, there is a way to define the factorial function for all numbers except the negative integers, by a domain-stretching exercise not unlike the one I just did.
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From page 148... ...
And if you look back at the statement of the Riemann hypothesis, you see that it concerns "all non-trivial zeros of the zeta function." Are we getting close? Alas, no, the negative even integers are in(lee zeros of the zeta function; but they are all, every one of them, trivial zeros.
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From page 149... ...
I can just integrate term by term, using the rules for integrating powers that I gave in Table 7-2. Here is the result (which was first obtained by Sir Isaac Newton)
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From page 150... ...
Convergent series fall into two categories: those that have this property, and those that don't. Series like this one, whose limit depends on the order in which they are summed, are called"conditionally convergent." Better-behaved series, those that converge to the same limit no matter how they are rearranged, are called "absolutely convergent." Most of the important series in analysis are absolutely convergent.
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