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Section24.7Exercises

1

Write down your answers to the three questions about the definition of Dirichlet series after Definition 24.3.1.

3

Learn more about the notion of zero density (recall Subsection 22.2.2). Then find other sets like \(P = \{\text{ primes }\}\) such that the sum of the reciprocals diverges, but the set has zero density in the integers.

4

Use Sage or other computational tools to conjecture the rate of growth of the function \begin{equation*}f(x)=\sum_{p\leq x}\frac{1}{p}\end{equation*} where \(p\) is of course prime. Hint: Typically one needs lumber to print a book, such as [C.3.5] (but don't peek there until you're really stuck!).

5

Recall \(\omega\) from Definition 23.3.3 and \(f(x)\) from the previous question. Confirm numerically that the average value to \(x\) (in the sense of Chapter 20 ) of \(\omega\) is about the same as the size of \(f(x)\). Give a reason why \(\sum_{p\leq x}\frac{1}{p}\) should be related to \(\sum_{n\leq x}\omega(n)\).