Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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21. The Error Term
21. The Error Term
Pages 327-349
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From page 327... ...
· The function Ican be written in terms of Riemann's zeta function I It follows that all the properties of the prime counting function fez are co(le(l, in some way, in the properties of A
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From page 328... ...
Expression 21-1 shows the result of that last inversion, the final and precise expression for I(x) in terms of the zeta function.
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From page 329... ...
The actual value of this area, the maximum value of the fourth term for any x we might ever be interested in, is, in fact,0.1400101011432869.... So the third and fourth terms taken together (and minding signs)
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From page 330... ...
The value of Lice) can, therefore, be looked up in a book of mathematical tables, or computed by any decent math software package like Maple or Mathematica.l26 Having thus disposed of the first, third, and fourth terms in Expression 21-1, I will focus on the second, ~Li( |
From page 331... ...
From the point of view of complex function theory, polynomial functions have a very interesting property. The (lomain of a polyno
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From page 332... ...
The log function is not an entire function, either: it has no value at argument zero. Riemann's zeta function, likewise, has no value at argument 1, and so is not an entire function.
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From page 333... ...
, with the sum being taken over an the non-trivial zeros of the zeta function. Now, to show the significance of this second term in Expression 21-1, and the problems it raises, I am going to take it apart.
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From page 334... ...
~ 2 3 4 . -3i 5 -4i · 9 20 , , , 14 Real ·16 FIGURE 21-2 The value plane for the function w = 20Z, showing the values of w for the first 20 non-trivial zeros of the zeta function.
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From page 335... ...
If you imagine our pal the argument ant walking north up the critical line in the argument plane with his function-ometer set to the function 20Z, his twin brother, the value ant, tracing out the corresponding values in the value plane, is walking round and round and round that circle. He is proceeding counterclockwise and by the time the argument ant has reached the first zeta zero, the value ant is nearly three-quarters of the way through his seventh circuit.
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From page 336... ...
For any nontrivial zero on the north half of the critical line, there is a corresponding one in the south half. That is, if ~ + 14.134725i is a zero of the zeta function, so must 2 - 14.134725ibe.
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From page 337... ...
For a fuller picture, including the southern half of the line, Figure 21 -4 shows, at the far left, a plane of complex numbers with the critical strip marked in from 2 - 15i to 2 + 15i. This is enough to show the first zero at 2 + 14.134725i, and also its complex conjugate at 2 - 14.134725i.
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From page 338... ...
, the value ant traversing the spiral in the value plane is getting closer and closer to Fez i at a rate inversely proportional to the argument ant's height. If the argument ant's height is T
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From page 339... ...
. To see how this process actually works out, and to get an insight into why Riemann calle(1 this secondary term the "periodic terms," let me work through the arithmetic for an actual value of x.
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From page 340... ...
After that, though, they get smaller as the values corresponding to the north half of the critical line spiral in toward Fez i.
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From page 341... ...
up and down, from positive to negative and back.~30 The reason for this is plain in Figure 21-3. The oscillatory quality of these secondary terms arises because, as Figure 21-3 shows, the function Lit win(ls the critical line roun and round in an ever-tighter spiral.
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From page 342... ...
The third term is log 2, which is 0.69314718055994.... The fourth term, that nuisance integral, deliv
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From page 343... ...
The Mobius function is zero, remember, for any number that is divisible by a perfect square like 4 or 9. Each of those 13 items has four terms: the principal term, the secondary term (which involves the zeros of the zeta function)
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is easy to work out, it comes to is . Because 10 is square-free and the product oftwo primes, its Mobius function ,u (10)
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From page 345... ...
The one for N= 19 is almost as big as the one for N= 6. Those secondary terms, the terms that involve zeros of the zeta function, are the wild cards in this calculation.
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From page 346... ...
· The top spiral an(1 the bottom spiral approach each other, "kiss" at some value of x between 100 an(1 1,000, an(1 thereafter overlap. (The spirals actually kiss when x= 399.6202933538....)
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From page 347... ...
THE ERROR TERM 347 imaginary 10i 5i ~ .
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From page 348... ...
The (liagram equivalent to Figure 21-3 has a far bigger hole in the mi(l(llethough still centered on Fez i and the spiral winds trillions of times between successive low-order zeros, scrambling their coordinates in the complex plane very effectively, the real parts of the values oscilIating between hugely negative an(1 hugely positive numbers. An(1 all this refers only to the first of the 639 table rows I nee(1 for computing adz (Bays-Hudson Number)
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From page 349... ...
, and the non-trivial zeros of the zeta function, which make up a large and, by Littlewood's result, sometimes dominant component of the difference between fez (x) and Like, that is, of the error term in the PNT.
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