Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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15. Big Oh and Möbius Mu
15. Big Oh and Möbius Mu
Pages 238-251
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From page 238... ...
She reported that his last murmured words were "Big oh of one...." Mathematicians tell this story with awed admiration. "Doing number theory to the very end A real mathematician!
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From page 239... ...
is forever trapped between two horizontal lines, one above the axis, one an equal distance below it. As I said, lots of functions are not big oh of one.
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From page 240... ...
If A is big oh of B then so is ten times A, a hundred times A, a million times A; so is one-tenth of A, one-hundredth of A, one-millionth of A
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From page 241... ...
This is x2, the squaring function. No matter how wide you make the pie wedge no matter how big the value of a the graph of x2 eventually crashes through the upper line.
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From page 242... ...
Notice also that, since big oh (loesn't care about multiples, the vertical scale is entirely arbitrary. It's the configuration that mattersthe shape of the bounding curves, and the fact that my function from some point on is forever trapped between them.
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From page 243... ...
The correspon(ling parabola shape in Figure 15-3 is a tad wi(ler, the (1ifference getting more and more noticeable as x goes out to infinity. If the Riemann Hypothesis is true, we have the best possible the tightest big-oh formula for the error term, O(4 logx)
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From page 244... ...
Furthermore, I showed in Chapter 5.iv that log x grows more slowly than any positive power of x, even the teeniest. This can be expressed using big-oh notation thus: For any number £, no matter how small, log x= 0(xE)
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From page 245... ...
"Oh," said Peter, "You should speak to my colleague Nick" (i.e., Nicholas Katz, also a professor at Princeton, though mainly an algebraic geometers. "Nick hates big oh.
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From page 246... ...
Once you have seen this, multiplying out an infinity of parentheses is a breeze. The answer is going to be a sum an infinite sum, of course of terms; and each term is got by plucking one number from
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From page 247... ...
Well, I think you can see that by plucking a 1 from every parenthesis except the nth, I am going to get a term equal to -l/pS, where p is the nth prime. So the infinite sum looks like Expression 15-3.
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From page 248... ...
One is the rule of signs, a minus times a minus gives a plus. The other is Power Rule 7, (x x ye n = xn X yen.
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From page 249... ...
that divide by some prime squared. Welcome to the Mobius function, named after the German mathematician and astronomer august Ferdinand Mobius (1790-1868~.83 It is universally referred to now by the Greek letter A, pronounced "mu," the Greek equivalent of "m."84 Here is a full (definition of the Mobius function band.
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From page 250... ...
increases, but nothing else is clear. Because of Expression 15-5' the behaviors of the ,u function and the M function (cumulative ,u ~ are intimately tied up with the zeta function and, therefore, with the Riemann Hypothesis.
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From page 251... ...
Theorem 15-2 If Theorem 15-2 is true, the Hypothesis is true; and if it is false, the Hypothesis is false. They are exactly equivalent theorems.
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