## Number Theory: Lecture 15

In which we meet the Riemann zeta function.

• Theorem 46 (Prime Number Theorem): $\displaystyle \pi(x) = \frac{x}{\log x}$.  That is, $\displaystyle \frac{\pi(x)}{x/\log x} \to 1$ as $x \to \infty$.
• Definition of the Riemann zeta function $\zeta$.
• Lemma 47: For $\Re(x) > 1$, the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges absolutely.  Moreover, the series converges uniformly for $\Re(s) \geq 1 + \delta$ for any $\delta >$, and so $\zeta$ is analytic in $\Re(s) > 1$.  We noted that $|n^s| = n^{\sigma}$ (where as usual $s = \sigma + it$) and then used our knowledge about $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{\sigma}}$.
• Proposition 48 (Euler product for $\zeta$): For $\Re(s) > 1$, we have $\zeta(s) = \prod_p (1-p^{-s})^{-1}$, where the product is over all primes $p$.  We used the idea that $\displaystyle \prod_p (1-p^{-s})^{-1} = \prod_p (1 + p^{-s} + p^{-2s} + \dotsb) = \sum_{n \geq 1}n^{-s}$, by the Fundamental Theorem of Arithmetic.
• Lemma 49: If $\Re(s) > 1$, then $\zeta(s) \neq 0$.  We carefully showed that for large enough $X$ we have $\displaystyle \zeta(s) \prod_{p\leq X} (1-p^{-s}) \geq \frac{1}{2}$.
• We defined the Gamma function $\Gamma$, and the completed $\zeta$ function $\Xi$, and saw the functional equation for $\zeta$.  We met the trivial zeros of $\zeta$, and identified the critical strip.
• Riemann Hypothesis: All the zeros of $\zeta$ in the critical strip lie on the line $\Re(s) = \frac{1}{2}$.
• Definition of the Möbius function $\mu$.
• Lemma 50: The Möbius function is multiplicative.  This followed immediately from the definition.
• We defined the Mertens function $\sum_{n \leq x} \mu(n)$, and noted (without proof) that the Riemann Hypothesis is equivalent to the bound $\sum_{n \leq x} \mu(n) = O_{\epsilon}(x^{\frac{1}{2} + \epsilon})$.

I handed out the third examples sheet.

#### Understanding today’s lecture

You could check that you understand the various arguments we used today (uniform convergence, etc.).

There are many books that give introductions to the Riemann zeta function.  Davenport’s Multiplicative Number Theory goes into more depth than some.  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 $\displaystyle \sum_{n \geq 1} \frac{\mu(n)}{n^s}$ in terms of $\zeta$?

Can you find an expression for $\displaystyle \frac{\zeta'(s)}{\zeta(s)}$ of the form $\displaystyle \sum_{n \geq 1} \frac{a_n}{n^s}$?  (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.