## 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.

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)?