*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.

#### Further reading

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.

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