*In which we encounter the Prime Number Theorem.*

- Theorem 50 (Prime Number Theorem):
*We have as . That is, as .*We saw some of the key ideas of the proof, but did not go into detail. - Definition of the
*von Mangoldt function*. - Lemma 51:
*If then .*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*corresponding to a sequence .

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 in Q6(i) should of course have been an . 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 , the number of primes less than . 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 (an exact expression, not an asymptotic formula)?

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