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}).

Further reading

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.

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One Response to “Number Theory: Lecture 15”

  1. Number Theory: Lecture 16 « Theorem of the week Says:

    […] Theorem of the week Expositions of interesting mathematical results « Number Theory: Lecture 15 […]

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