In which we study the local factors in more detail.
- Lemma 16: The sum is multiplicative, in the sense that if and are coprime natural numbers then
for any coprime to and any coprime to . We quickly proved this, essentially using the Chinese Remainder Theorem.
- Lemma 17: If and are coprime, then . (That is, is multiplicative.) This was easy using Lemma 16.
- Corollary 18: The are the local factors for this problem, in the sense that . This follows immediately from Lemma 17 and the absolute convergence of the series for .
- Lemma 19: (i) Let be an odd prime, and write where and are coprime. Let be an integer not divisible by . Suppose that there is some with . Then for each there is a solution to .
(ii) Write where is odd. Let be an odd integer. Suppose that there is some with . Then for each there is a solution to . We proved this by considering the structure of the multiplicative group modulo . One could also use a suitable form of Hensel’s Lemma.
- Lemma 20: (i) Let be an odd prime, and write where and are coprime. Suppose that we have such that . Suppose moreover that . Then for any there are at least solutions to the congruence .
(ii) Write where is odd. Suppose that we have such that . Suppose moreover that is odd. Then for any there are at least solutions to the congruence . This was easy with the help of Lemma 19.
Preparation for Lecture 9
Our next task is to show that there is a solution to (for suitable ), for large enough . You could try to prove this yourself ahead of the lecture on Wednesday.