## Number Theory: Lecture 16

In which we see an outline of a proof of the Prime Number Theorem.

• Definition of the von Mangoldt function.
• Lemma 51: If $\Re(s) > 1$, then $\displaystyle \frac{\zeta'(s)}{\zeta(s)} = - \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$.  The left-hand side is the logarithmic derivative of $\zeta$.  We obtained the right-hand side by taking the logarithm of the Euler product for $\zeta$ and differentiating.
• Definition of a Dirichlet series.

#### Understanding today’s lecture

What other interesting Dirichlet series can you come up with?  What are the Dirichlet series corresponding to the arithmetic functions we’ve met so far?  Can you find any connections between them?

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.

I mentioned the work being done on connections between zeros of the $\zeta$ function and random matrix theory.  Jon Keating, who has done a lot of work in the area, has a survey paper about this.

Mathematicians who were mentioned today but don’t appear above so need links to their biographies: Hadamard, de la Vallée Poussin, Selberg.

#### 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)?