The best way to understand mathematics is to do mathematics. In keeping with this principle, this week's discussion offers you a series of questions and problems that further develop the ideas presented in this blog post. If you'd like to discuss your ideas and solutions with your Brilliant fellows, please post them below. I will also comment on them wherever appropriate.
Calculate , where is the number of positive integer divisors of .
Does the series converge? If so, what is its sum?
For an integer , let denote the number of all odd positive integer divisors of . Find the sum .
Does the series converge?
For an integer , let's define if is divisible by and otherwise. Does the series converge? If so, what is the sum?
Same question for the series .
Prove that the series converges conditionally (but not absolutely) for any real number , and converges absolutely for any .
Investigate the convergence of the last series for .
What is the analogue of Dirichlet's asymptotic formula for a sum of the form , where are fixed integers and are relatively prime?