Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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3. The Prime Number Theorem
3. The Prime Number Theorem
Pages 32-47
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From page 32... ...
A prime number is one with no proper factors. Here are all the prime numbers up to 1,000.
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From page 33... ...
Including 1 in the primes, however, is a major nuisance, and modern mathematicians just don't, by common agreement. (The last mathematician of any importance who did seems to have been Henri Lebesgue, in 1899.)
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From page 34... ...
So either it doesn't have any proper factors and therefore is itself a prime bigger than N or its smallest proper factor is some number bigger than N Since any number's smallest proper factor is bound to be a prime if it wasn't, it could be factored down into something smaller this proves the result.
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From page 35... ...
TABLE 3- 1 N How many primes less than FE 1,000 1,000,000 1,000,000,000 1,000,000,000,000 1,000,000,000,000,000 1,000,000,000,000,000,000 168 78,498 50,847,534 37,607,912,018 29,844,570,422,669 24,739,954,287,740,860 That's nice, but not actually terribly informative. Yes, the primes sure do thin out.
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From page 36... ...
If you try, the game stops and everyone goes back to his last legal position.) For another example, consider the function whose rule is "count the number of factors the argument has." You find that 28 has six factors (I'm including trivial factors here)
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From page 37... ...
for a particular function by working intimately with it for a long time, observing all its features and peculiarities. A table or a graph rarely encompasses the whole thing.
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From page 38... ...
The Greek alphabet has only 24 letters and by the time mathematicians got round to giving this function a symbol (the person responsible in this case is Edmund Landau, in 1909 see Chapter 14.iv) , all 24 had been pretty much used up and they had to start recycling them.
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From page 39... ...
From the point of view I have adopted here functions illustrated by two-column tables, like Table 3-1 I can give you a loose definition of an exponential function as follows. If you pick your arguments so that they go up by regular addition from row to row, and then apply the function rule to them, and if it turns out that the resulting values go up by regular multiplication from row to row, you are looking at an exponential function.
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From page 40... ...
There is one exponential function mathematicians prefer above all others. If you were to take a guess at it, you might suppose it is the one in which the multiplier is 2 the simplest number to multiply by, after all.
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From page 41... ...
I can't explain the importance of e without going into calculus, though, and I have sworn a solemn oath to explain the Riemann Hypothesis with the utter minimum of calculus. I am, therefore, just going to beg you to take on faith that e is a really, real/l/y important number, and that no other exponential function can hold a candle to this one, the function eN.
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From page 42... ...
N N2 -3 -2 o 1 2 3 9 4 o 4 9 (I'm assuming you remember the rule of signsi2 here, so that-3 times -3 is 9, not -9.) Now, if you flip columns, you get the inverse function.
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From page 43... ...
You can cheerfully invert it to give you a function that, when you pick arguments going up by multiplication, gives you values going up by a(l(lition. Of course, as with exponential functions, there is a whole family of inverse functions, depending on the multiplier; and as with the exponential function, mathematicians much, much prefer the one that goes up in additions of 1 when the arguments go up in multiples of e.
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From page 44... ...
In the mathematical topics relevant to this book relevant, that s, to the Riemann Hypothesis the log function is everywhere. I shall have much more to say about it in Chapters 5 and 7, and it will play a starring role when I actuaPy turn the Golden Key in Chapter 19.
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From page 45... ...
It is a very important result; so important that it is called "the Prime Number Theorem." Not"a prime number theorem." This is "the Prime Number Theorem." Note the capital letters, which I shall use when referring to the theorem. Very often, in fact, when the context is sufficiently plain, number theorists simply write "PNT," a practice I shall follow in this book.
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From page 46... ...
It predicts, for example, that the trillionth prime will be 27,631,021,115,929; in fact, the trillionth prime is 30,019,171,804,121, an 8 percent error. Percent errors at a thousand, a million, and a billion are 13,10, and 9.
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From page 47... ...
THE PRIME NUMBER THEOREM 47 Not only are these consequences of the PNT; it is also a consequence of them. If you could mathematically prove the truth of either, the PNT would follow.
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