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 , and , .
- We noted that if and are not coprime, then this result clearly does not hold.
- Theorem 43 (Dirichlet): Let be an integer greater than 1, and let be coprime to . Then there are infinitely many primes congruent to modulo . We did not prove this! But we saw a small hint of how one can prove it.
- Proposition 44: For any , we have , where the sum is over all primes . We proved this by first showing that , and then that is bounded above by a constant (1/2, as it happens).
- Corollary 45: There are infinitely many primes.
- Proposition 46: There is a constant such that . We proved this using an idea of Erdős, namely to write each as , where the are the primes up to and each is 0 or 1, and then to count the number of such expressions.
Wikipedia has a page with a couple of ways of thinking about . 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 function, which is defined on natural numbers. If , where the are distinct primes, then . If is not square-free (if it is not the product of distinct primes), then . And, just to be clear, . Here are a couple of questions to get you started thinking about this function.
- Is a multiplicative function?
- Can you compute the sums for various ? (The sum is over all divisors of .)
- Can you make any conjectures about the approximate size of the sum ? You might like to use a computer to help you collect some data, in addition to thinking about the function .
- You might like to try to relate and the Euler totient function that we saw earlier in the course.