NTIC Exercises

Exercises 24.7 Exercises

1.

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

2.

Prove Theorem 24.5.4 in full generality, following that of Fact 24.5.5. (This is a good technical exercise in convergence.)

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3.

Look up, or prove from scratch, that the ‘alternating harmonic series’ $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k}$ is convergent, but not absolutely convergent. Look up, or prove from scratch, the value of this series; then find a rearrangement of it that sums to precisely half the usual value. (Extra credit if you do so without referencing anything connected to the university IUPUI.)

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The sum of the reciprocals of all primes is a very nuanced thing; here are some additional exercises about it.

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4.

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

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5.

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 [E.4.5] (but don't peek there until you're really stuck!).

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6.

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 $f(x)\text{.}$ Give a reason why $\sum_{p\leq x}\frac{1}{p}$ should be related to $\sum_{n\leq x}\omega(n)\text{.}$

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7.

Find an exercise about averages of arithmetic functions, Dirichlet series, or Euler products in [E.4.6, Chapters 3 and 11] and create a Sage cell to verify the result computationally. Then do the actual exercise, and report back comparing the two experiences.

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8.

Following [E.7.35], let a point $r,s$ be $b$ -visible from the origin ( $b$ a positive integer) if it lies on the graph of some $y=ax^b$ for $a\in \mathbb{Q}$ and there is no other lattice point between that point and the origin on the curve. Theorem 1 of their paper is that the proportion of $b$ -visible points is $\frac{1}{\zeta(b+1)}\text{.}$ Verify this experimentally using graph paper or a computer for $b=2\text{.}$