## Number Theory: Lecture 15

In which we meet the Riemann zeta function and start to explore what it can tell us about prime numbers.

• Definition of the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$.
• Lemma 47: If $\Re(s) > 1$, then $\sum_{n=1}^{\infty} \frac{1}{n^s}$ converges absolutely.  Moreover, for any $\delta > 0$ the series converges uniformly for $\Re(s) \geq 1 + \delta$, and so is analytic on $\Re(s) > 1$.  The important point to remember here is that $s$ is complex.  Writing $s = \sigma + it$ (as is standard, if bizarre), we have $|n^{-s}| = n^{-\sigma}$.
• Proposition 48 (Euler product for $\zeta$): If $\Re(s) > 1$, then $\zeta(s) = \prod_p (1-p^{-s})^{-1}$, where the product is over all primes $p$.  We proved this by considering the ‘partial product’ $\prod_{p \leq N} (1-p^{-s})^{-1}$.  Note that this result essentially records the fundamental theorem of arithmetic.  This Euler product is crucial for linking the zeta function to the primes.
• Lemma 49: If $\Re(s) > 1$, then $\zeta(s) \neq 0$.  We proved this using the Euler product, being careful about the infinite product (we showed that for a suitable $X$, we have $\prod_{p > X} (1-p^{-s})^{-1} > \frac{1}{2}$, and could then check individual factors $(1-p^{-s})^{-1}$ for primes $p \leq X$.
• We noted (without proof) some properties of $\zeta$, such as the continuation to a meromorphic function on the complex plane, and the functional equation.  We saw the Riemann Hypothesis, which asserts that all the non-trivial zeros of $\zeta$ lie on the line $\Re(s) = \frac{1}{2}$.
• Definition of the Möbius function $\mu$.  Exercise: show that $\mu$ is a multiplicative function.
• We noted (without proof) that the Riemann Hypothesis is equivalent to the bound $\sum_{n \leq x} \mu(n) = O_{\epsilon}(x^{\frac{1}{2} + \epsilon})$.

There are many books that give introductions to the Riemann zeta function.  Davenport’s Multiplicative Number Theory goes into more depth than some.  Ben Green has some online lecture notes from a Part III course a few years ago.  The book Analytic Number Theory by Iwaniec and Kowalski tells you more than you realised you wanted to know about the zeta function and much else besides.

#### Preparation for Lecture 16

What is the relationship between $\mu$ and $\zeta$?  Hint: can you write $\sum_{n \geq 1} \frac{\mu(n)}{n^s}$ in terms of $\zeta$?

Can you find an expression for $\frac{\zeta'(s)}{\zeta(s)}$ of the form $\sum_{n \geq 1} \frac{a_n}{n^s}$?  (This latter series is called the Dirichlet series for the sequence $a_n$.)  The values of $a_n$ will be important when we come to think about the Prime Number Theorem.