Number Theory: Lecture 16

In which we encounter the Prime Number Theorem.

  • Theorem 50 (Prime Number Theorem): We have \pi(x) \sim \frac{x}{\log x} as x \to \infty.  That is, \pi(x)/(x/\log x) \to 1 as x \to \infty.  We saw some of the key ideas of the proof, but did not go into detail.
  • Definition of the von Mangoldt function \Lambda.
  • Lemma 51: If \Re(s) > 1 then \frac{\zeta'(s)}{\zeta(s)} = - \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}.  The left-hand side is the logarithmic derivative of the zeta function.  We saw that we could obtain the right-hand side by taking the logarithm of the Euler product and then differentiating.
  • Definition of the Dirichlet series \sum_{n=1}^{\infty} \frac{a_n}{n^s} corresponding to a sequence (a_n).

I gave out the third examples sheet, which is now available on the course page.  I am grateful to the eagle-eyed student who spotted a typo before the end of the lecture: the first N in Q6(i) should of course have been an N!.  I have corrected this in the online version.  Apologies for any confusion this may have caused.

Further reading

The suggestions I made for the last lecture are just as valid for this lecture.  There’s some nice material on the distribution of the primes in Topics in the Theory of Numbers, by Erdős and Surányi.  There’s some discussion of an elementary proof of the Prime Number Theorem in An Introduction to the Theory of Numbers by Hardy and Wright (at least in the later editions — there was no known elementary proof when the first edition was written!).   Terry Tao has a nice blog post discussing a number of aspects of the distribution of prime numbers.

Preparation for Lecture 17

We’re going to move on to another formula for \pi(x), the number of primes less than x.  How would you compute the number of primes less than 100, or less than 1000 (without a computer, and without just listing all the primes!).  Can you generalise your ideas to find an expression for \pi(x) (an exact expression, not an asymptotic formula)?

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