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 $\sum_{n \leq x} \mu(n) = O_{\epsilon}(x^{\frac{1}{2} + \epsilon})$ .

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.

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One Response to “Number Theory: Lecture 15”

Number Theory: Lecture 16 « Theorem of the week Says:
November 11, 2011 at 9:18 pm
[...] Theorem of the week Expositions of interesting mathematical results « Number Theory: Lecture 15 [...]

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