4400pnt - THE PRIME NUMBER THEOREM AND THE RIEMANN...

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THE PRIME NUMBER THEOREM AND THE RIEMANN HYPOTHESIS PETE L. CLARK 1. Some history of the prime number theorem Recall we have defined, for positive real x , π ( x ) = # { primes p x } . The following is probably the single most important result in number theory. Theorem 1. (Prime Number Theorem) We have π ( x ) x log x ; i.e., lim x →∞ π ( x ) log x x = 1 . 1.1. Gauss at 15. The prime number theorem (affectionately called “PNT”) was apparently first conjectured in the late 18th century, by Legendre and Gauss (inde- pendently). In particular, Gauss conjectured an equivalent – but more appealing – form of the PNT in 1792, at the age of 15 (!!!). Namely, he looked at the frequency of primes in intervals of lengths 1000: ∆( x ) = π ( x ) π ( x 1000) 1000 . Computing by hand, Gauss observed that ∆( x ) seemed to tend to 0, however very slowly. To see how slowly he computed the reciprocal, and found 1 ∆( x ) log x, meaning that ∆( x ) 1 log x . Evidently 15 year old Gauss knew both differential and integral calculus, because he realized that ∆( x ) was a slope of the secant line to the graph of y = π ( x ). When x is large, this suggests that the slope of the tangent line to π ( x ) is close to 1 log x , and hence he guessed that the function Li( x ) := x 2 dt log t was a good approximation to π ( x ). Proposition 2. We have Li( x ) x log x . 1
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2 PETE L. CLARK Proof: A calculus exercise (L’Hˆopital’s rule!). Thus PNT is equivalent to π ( x ) Li( x ). The function Li( x ) – called the logarith- mic integral – is not elementary, but has a simple enough power series expansion (see for yourself). Nowadays we have lots of data, and one can see that the error | π ( x ) Li( x ) | is in general much smaller than | π ( x ) x log x | , so the dilogarithm gives a “better” asymptotic expansion. (How good? Read on.) 1.2. A partial result. As far as I know, there was no real progress for more than fifty years, until the Russian mathematician Pafnuty Chebyshev proved the following two impressive results. Theorem 3. (Chebyshev, 1848, 1850) a) There exist explicitly computable positive constants C 1 , C 2 such that for all x , C 1 x log x < π ( x ) < C 2 x log x . b) If lim x →∞ π ( x ) x/ (log x ) exists, it necessarily equals 1 . Remarks: (i) For instance, one version of the proof gives C 1 = 0 . 92 and C 2 = 1 . 7 . (But I don’t know what values Chebyshev himself derived.) (ii) The first part shows that π ( x ) is of “order of magnitude” x log x , and the second shows that if it is “regular enough” to have an asymptotic value at all, then it must be asymptotic to x log x . Thus the additional trouble in proving PNT is establishing this regularity in the distribution of the primes, a quite subtle matter. (We have seen that other arithmetical functions, like φ and d are far less regular than this – their upper and lower orders differ by more than a multiplicative constant, so the fact that this regularity should exist for π ( x ) is by no means assured.) (iii) Chebyshev’s proof is quite elementary: it uses less machinery than some of the other topics in this course. However we will not give the time to prove it here: blame it on your instructor’s failure to “understand” the proof.
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