Number Theory: Lecture 14

In which we start to explore the distribution of the primes.

  • We reminded ourselves of some examples of results that we have previously seen of the form “There are infinitely many primes congruent to a modulo n”, such as when a = 1, n = 4 and a = 3, n = 4.
  • We noted that if a and n are not coprime, then this result clearly does not hold.
  • Theorem 43 (Dirichlet): Let n be an integer greater than 1, and let a be coprime to n.  Then there are infinitely many primes congruent to a modulo n.  We did not prove this!  But we saw a small hint of how one can prove it.
  • Proposition 44: For any x \geq 10, we have \sum_{p \leq x} \frac{1}{p} \geq \log \log x - \frac{1}{2}, where the sum is over all primes p \leq x.  We proved this by first showing that \prod_{p \leq x} (1-\frac{1}{p})^{-1} \geq \log x, and then that \log \left(\prod_{p \leq x} (1 - \frac{1}{p})^{-1} \right) - \sum_{p \leq x} \frac{1}{p} is bounded above by a constant (1/2, as it happens).
  • Corollary 45: There are infinitely many primes.
  • Proposition 46: There is a constant c > 0 such that \pi(x) \geq c \log x.  We proved this using an idea of Erdős, namely to write each y \leq x as y = m^2 p_1^{\alpha_1} \dotsm p_k^{\alpha_k}, where the p_i are the primes up to x and each \alpha_i is 0 or 1, and then to count the number of such expressions.

Further reading

Wikipedia has a page with a couple of ways of thinking about \sum_{p \leq x} \frac{1}{p}.  Hardy and Wright (An introduction to the theory of numbers) tackles this problem and some related results (see section 22.7).  Erdős and Surányi (Topics in the theory of numbers) give a nice presentation of this material.  Davenport (Multiplicative Number Theory) discusses the sum of the reciprocals of the primes, and also gives a proof of Dirichlet’s theorem on primes in arithmetic progressions.  (Note that this is a different book by Davenport, not the one that I’ve recommended in previous posts!)

On a separate note, I spotted a new paper on the arXiv this morning about positive definite binary quadratic forms.   I haven’t looked beyond the abstract, but it does demonstrate that this is still very much a topic of current research.

Preparation for Lecture 15

Next time we shall move on to the Riemann zeta function, and how it is related to the distribution of the primes.   I might mention the Möbius \mu function, which is defined on natural numbers.  If n = p_1 p_2 ... p_k, where the p_i are distinct primes, then \mu(n) = (-1)^k.  If n is not square-free (if it is not the product of distinct primes), then \mu(n) = 0.  And, just to be clear, \mu(1) = 1. Here are a couple of questions to get you started thinking about this function.

  • Is \mu a multiplicative function?
  • Can you compute the sums \sum_{d|n} \mu(d) for various n?  (The sum is over all divisors d of n.)
  • Can you make any conjectures about the approximate size of the sum \sum_{n \leq x} \mu(n)?  You might like to use a computer to help you collect some data, in addition to thinking about the function \mu.
  • You might like to try to relate \mu and the Euler totient function \phi that we saw earlier in the course.
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