An extended hat puzzle

Shortly after hearing about the hat puzzle I wrote about last month I came across an interesting extension of the problem, which replaces the 100 wizards with an infinite number of wizards:

A countably infinite number of wizards are each given a red or blue hat with 50% probability. Each wizard can see everyone’s hat except their own. The wizards have to guess the colour of their hat without communicating in any way, but will be allowed to devise a strategy to coordinate their guesses beforehand. How can they ensure that only a finite number of them guess incorrectly? You may assume the axiom of choice.

This seems paradoxical since somehow knowing about other wizard’s hats—which are chosen independently from a wizard’s own hat—allows each wizard to conclude that they will almost surely guess their hat colour correctly.

Volume of a hypersphere in the 1-norm

Previously I derived the volume of a hypersphere in $n$ dimensions. A hypersphere with radius $R$ consists of the set of points $\newcommand{\x}{\mathbf{x}}\newcommand{\R}{\mathbb{R}}\x=(x_1,\dotsc,x_n)\in\R^n$ for which

\[ \lVert\x\rVert \leq R , \]

where $\lVert\x\rVert$ denotes the usual Euclidean norm (also known as the 2-norm),

\[ \lVert\x\rVert := \sqrt{x_1^2+\dotsb+x_n^2} . \]

Today, I’d like to consider the problem of computing the volume of an $n$-dimensional hyphersphere in the 1-norm (also known as the Manhattan distance or taxicab norm), which is defined by

\[ \lVert\x\rVert_1 := \lvert x_1\rvert+\dotsb+\lvert x_n\rvert . \]

The volume of the 1-norm hypersphere is given by the expression

\[ V_n(R) := \frac{(2R)^n}{n!} , \]

as we will show by induction on $n$. In the base case $n=1$ one has

\[ \newcommand{\d}{\,\mathrm{d}} V_1(R) = \int_{-R}^R\d x_1 = 2R , \]

as required. Now suppose that the formula holds in dimension $n-1$. Then we have

V_n(R) &= \int\limits_{\lvert x_1\rvert\leq R}\;\int\limits_{\lvert x_1\rvert+\lvert x_2\rvert\leq R}\dotsi\int\limits_{\lvert x_1\rvert+\dotsb+\lvert x_n\rvert\leq R}\d x_n\dotsm\d x_1 \\
&= \int\limits_{\lvert x_1\rvert\leq R}\;\int\limits_{\lvert x_2\rvert\leq R-\lvert x_1\rvert}\dotsi\int\limits_{\lvert x_2\rvert\dotsb+\lvert x_n\rvert\leq R-\lvert x_1\rvert}\d x_n\dotsm\d x_1 \\
&= \int\limits_{\lvert x_1\rvert\leq R} V_{n-1}\bigl(R-\lvert x_1\rvert\bigr) \d x_1 \\
&= \int_{-R}^R \frac{2^{n-1}(R-\lvert x_1\rvert)^{n-1}}{(n-1)!} \d x_1 \\
&= 2\int_{0}^R \frac{2^{n-1}(R-x_1)^{n-1}}{(n-1)!} \d x_1 \\
&= \frac{2^n}{(n-1)!}\biggl[-\frac{1}{n}(R-x_1)^n\biggr]_0^R \\
&= \frac{(2R)^n}{n!}

By induction, the formula holds for all positive integers $n$.

Harmonic series variant solution

Last year I asked if the series $\sum_{n=1}^\infty\cos n/n$ converges or diverges. Although the series looks rather simple, the standard convergence tests learned in Calculus 1 do not apply directly. However, there is a trick which allows one to rewrite the series into a form where the standard tests can be used.

The trick is to use summation by parts, the discrete analog of integration by parts. Although less well-known, summation by parts does have its uses; for example, it was the method of proving Abel’s summation formula which has come up before.

To simplify the exposition, it is convenient to define a function for the partial sums of the series’ numerators:

\[ C(m) := \sum_{n=1}^m \cos n \]

Now, using summation by parts the series may be rewritten so that

\[ \sum_{n=1}^m\frac{\cos n}{n} = \frac{C(m)}{m} + \sum_{n=1}^{m-1} C(n)\Bigl(\frac{1}{n}-\frac{1}{n+1}\Bigr) . \tag{1} \]

The key observation to make now is that $C(m)$ is bounded. This can be seen by using the complex exponential expression of the cosine, and summing a geometric series to find a closed-form expression for $C(m)$, as follows:

C(m) &= \sum_{n=1}^m\frac{e^{in}+e^{-in}}{2} \\
&= \frac{e^i}{2}\Bigl(\frac{e^{im}-1}{e^i-1}\Bigr)+\frac{e^{-i}}{2}\Bigl(\frac{e^{-im}-1}{e^{-i}-1}\Bigr) \\
&= \frac{(e^{im}-1)(1-e^i)+(e^{-im}-1)(1-e^{-i})}{2(e^i-1)(e^{-i}-1)} \\
&= \frac{(e^{im}+e^{-im})-(e^{i(m+1)}+e^{-i(m+1)})}{2(2-(e^i+e^{-i}))}-\frac{1}{2} \\
&= \frac{\cos m-\cos(m+1)}{2(1-\cos1)}-\frac{1}{2}

By the triangle inequality, this has an absolute upper bound of

\[ \lvert C(m)\rvert \leq \frac{1}{1-\cos1}+\frac{1}{2}<3 . \]

From this it follows that $C(m)/m\to0$ as $m\to\infty$, and after taking the limit (1) becomes

\[ \sum_{n=1}^\infty\frac{\cos n}{n} = \sum_{n=1}^\infty{}C(n)\Bigl(\frac{1}{n}-\frac{1}{n+1}\Bigr) , \tag{2} \]

and we can use the absolute comparison test on the right sum. Using the fact that

\[ \frac{1}{n}-\frac{1}{n+1} = \frac{1}{n^2+n} \leq \frac{1}{n^2} \]

we have $\bigl\lvert C(n)(\frac{1}{n}-\frac{1}{n+1})\bigr\rvert<3/n^2$, and $\sum_{n=1}^\infty 3/n^2=3\zeta(2)$ is absolutely convergent, so by the comparison test the right sum in (2) is absolutely convergent as well. Thus the left side of (2) must converge, so $\sum_{n=1}^\infty\cos n/n$ converges.

Minkowski’s Theorem

Minkowski’s theorem says that if $S$ is a convex set in $\newcommand{\R}{\mathbb{R}}\R^n$ which is symmetric about the origin and has volume $V(S)>2^n$ (or if $S$ is compact and $V(S)\geq2^n$) then $S$ contains some nonzero point of $\newcommand{\x}{\mathbf{x}}\newcommand{\Z}{\mathbb{Z}}\Z^n$.

Minkowski’s theorem can be seen as a simple consequence of Blichfeldt’s theorem. In particular, consider applying its statement to the set $S/2:=\{\,s/2:s\in S\,\}$. Since \[V(S/2)=V(S)/2^n>1\] (or if $S$ is compact, $V(S/2)\geq1$) the theorem says that there exist distinct $\x_1$, $\x_2\in S/2$ with $\x_1-\x_2\in\Z^n$. Say that these have the form $\newcommand{\s}{\mathbf{s}}\x_1=\s_1/2$ and $\x_2=\s_2/2$ where $\s_1$, $\s_2\in S$.

Since $S$ is symmetric about the origin, it follows that $-\s_2\in S$. Additionally, since $S$ is convex, it follows that the midpoint of $\s_1$ and $-\s_2$ is also in $S$. But this midpoint $(\s_1-\s_2)/2=\x_1-\x_2$ is also in $\Z^n$. Since $\x_1\neq\x_2$ this point is nonzero, as required.

Using the form of Blichfeldt’s theorem applied to general lattices $L$ of dimension $n$, one finds that if $S$ is a convex and symmetric set in $\R^n$ with volume $V(S)>2^n\det(L)$ (or if $S$ is compact and $V(S)\geq2^n\det(L)$) then $S$ contains a nonzero point of $L$.

Blichfeldt’s Theorem

The following theorem was proven by Blichfeldt1 in 1914:

Let $P$ be a set of points in $\newcommand{\R}{\mathbb{R}}\R^n$ which are invariant under translation by vectors in $\newcommand{\Z}{\mathbb{Z}}\Z^n$, and say that $[0,1)^n$ contains $N$ points of $P$. If $S$ is a set in $\R^n$ with volume $V(S)$ then there is some translation $\newcommand{\x}{\mathbf{x}}S+\x$ which contains at least $N\cdot V(S)$ points of $P$. Additionally, if $S$ is compact then there is some translation of $S$ which contains more than $N\cdot V(S)$ points of $P$.

To show this, let $\newcommand{\p}{\mathbf{p}}\newcommand{\c}{{,}~}\p_1\c\dotsc\c\p_N$ be the points of $P$ contained in $[0,1)^n$. Since $P$ is invariant under translation by $\Z^n$, every $\p\in P$ is equivalent to some $\p_i$ modulo $\Z^n$. Let $\nu_i(S)$ denote the number of points in $P\cap S$ congruent to $\p_i$, so

\[ \newcommand{\y}{\mathbf{y}}\nu_i(S) = \sum_{\y\in\Z^n}\chi_S(\p_i+\y) \]

where $\chi_S$ is the characteristic function of $S$. Then

\int_{[0,1)^n}\nu_i(S+\x)\d\x &= \int_{[0,1)^n}\sum_{\y\in\Z^n}\chi_{S+\x}(\p_i+\y)\d\x \\
&= \int_{[0,1)^n}\sum_{\y\in\Z^n}\chi_S(\p_i+\y-\x)\d\x \\
&= \int_{(-1,0]^n}\sum_{\y\in\Z^n}\chi_S(\p_i+\y+\x)\d\x \\
&= \sum_{\y\in\Z^n}\int_{(-1,0]^n+\p_i+\y}\chi_S(\x)\d\x \\
&= \int_{\R^n}\chi_S(\x)\d\x \\
&= V(S)

where the penultimate step follows since taking the set $(-1,0]^n+\p_i+\y$ for each $\y\in\Z^n$ will produce a tiling of all of $\R^n$ with no overlap.

Letting $\nu(S):=\lvert P\cap S\rvert=\sum_{i=1}^N\nu_i(S)$ and summing the above over $1\leq i\leq N$, we find that

\[ \int_{[0,1)^n}\nu(S+\x)\d\x = N\cdot V(S) . \]

By an averaging argument, there must be some $\x\in[0,1)^n$ such that $\nu(S+\x)/V([0,1)^n)\geq N\cdot V(S)$, i.e., the translation $S+\x$ contains at least $N\cdot V(S)$ points of $P$, as required.

Next, we consider the case where $S$ is compact (i.e., closed and bounded). Since $\nu(S+\x)$ is an integer, if $N\cdot V(S)$ is not an integer then it immediately follows that $\nu(S+\x)>N\cdot V(S)$. So suppose that $h:=N\cdot V(S)$ is an integer. Let $S_k:=(1+\frac{1}{k})S$, so that applying the above result on $S_k$ for each $k\geq1$ we have that there exists some $\x_k\in[0,1)^n$ such that $\nu(S_k+\x_k)\geq N\cdot V(S_k)>h$.

Since the $\x_k$ lie in $[0,1]^n$ which is compact, there must be some subsequence $\x_{k_j}$ which converges to a point $\x\in[0,1]^n$. By construction, each $S_{k_j}+\x_{k_j}$ contains at least $h+1$ points of $P$. Since $\bigcup_j S_{k_j}+\x_{k_j}$ is bounded, it contains a finite number of points of $P$. In particular, there are a finite number of ways of choosing $h+1$ points of $P$ from $\bigcup_j S_{k_j}+\x_{k_j}$. By the pigeonhole principle, there must be some choice $\p_1\c\dotsc\c\p_{h+1}\in P$ which appear infinitely often in the sets $S_{k_j}+\x_{k_j}$. Let $k’_j$ index a subsequence of these sets so that $\p_i\in S_{k’_j}+\x_{k’_j}$ for all $i$ and $j$.

The points $\p_i$ must in fact appear in $S+\x$; otherwise there would be some $\p_i$ with positive distance to $S+\x$, but the distance of the farthest point of $S_{k’_j}+\x_{k’_j}$ to $S+\x$ goes to $0$ as $j\to\infty$, contradicting the supposed positive distance that $\p_i\in S_{k’_j}+\x_{k’_j}$ is away from $S+\x$. As required, we have found some translation $S+\x$ which contains more than $h=N\cdot V(S)$ points of $P$.

Alternate statement

Sometimes Blichfeldt’s theorem is stated in the following form:

Let $S$ be a set in $\R^n$ with volume $V(S)>m$ for some $m\in\Z$ (if $S$ is compact we only require $V(S)\geq m$). Then $S$ contains $\x_1\c\dotsc\c\x_{m+1}$ distinct points whose differences $\x_i-\x_j$ all lie in $\Z^n$.

This is a simple corollary of what we’ve already shown. Take $P:=\Z^n$, so that $N=1$. The theorem tells us there is some $\x$ such that $S+\x$ contains at least $V(S)\geq m+1$ (or if $S$ is compact, more than $V(S)\geq m$) points of $P$. Thus we have there exist $\p_i\in P$ and $\x_i\in S$ with $\p_i=\x_i+\x$ for $1\leq i\leq m+1$. Then

\[ \x_i-\x_j = \p_i-\p_j \in \Z^n , \]

as required.


Finally, Blichfeldt’s theorem can be generalized to apply to arbitrary lattice points, instead of just $\Z^n$. One possible way of stating it would be:

Let $L$ be a lattice in $\R^n$ with volume $\det(L)$, and $S$ be a set in $\R^n$ with volume $V(S)>m\det(L)$ for some $m\in\Z$ (if $S$ is compact we only require $V(S)\geq m\det(L)$). Then $S$ contains $\x_1\c\dotsc\c\x_{m+1}$ distinct points whose differences $\x_i-\x_j$ all lie in $L$.

The same argumentation as above suffices to show this, except that we replace $[0,1)^n$ (a fundamental region for $\Z^n$) in the argument with a fundamental region for $L$.

Two characterizations of a lattice

A lattice is a discrete additive subgroup of $\newcommand{\R}{\mathbb{R}}\R^n$; for the purposes of this post we’ll restrict ourselves to full rank lattices, i.e., those which span the entire vector space $\R^n$. An alternate way of defining a lattice is to consider it as the integer span of a collection of linearly independent vectors $\newcommand{\b}{\mathbf{b}}\newcommand{\c}{{,}~}\b_1\c\dotsc\c\b_n\in\R^n$, known as a basis of $L$. That is, $L$ consists of the collection of points

\[ \newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\x}{\mathbf{x}}\newcommand{\Z}{\mathbb{Z}} \biggl\{\,\sum_{i=1}^n x_i\b_i : x_i\in\Z \,\biggr\} . \]

The goal of this post is to show that these two ways of defining a lattice are equivalent. It is more or less immediate that a lattice $L$ in the second sense is an additive subgroup which spans $\R^n$; it is also discrete since one can show that1

\[ \newcommand{\0}{\mathbf{0}} \norm{\x} \geq \min_{1\leq i\leq n}\norm{\b_i^*} \]

for all $\x\in L$ and $\x\neq\0$, where $\b_i^*$ is the Gram–Schmidt orthogonalization of $\b_i$. Thus $\0$ is an isolated point of $L$, and it follows that every point of $L$ must be isolated; a nonisolated point would imply (using a suitable translation) that $\0$ is nonisolated.

The harder direction is to show that a lattice in the first sense is also a lattice in the second sense; the remainder of the post is devoted to this. Let $L$ be a discrete additive subgroup of $\R^n$ containing $n$ linearly independent vectors.

In the case $n:=1$, since $L$ spans $\R$ it must contain $\pm a\neq0$. Since it also contains $0$ and is discrete, there must be a minimal $b>0$ with $b\in L$. Then as required we have $L= \{\, xb : x\in\Z \,\}$; if there was any point $c=xb\in L$ for $x\notin\Z$ then $c-\lfloor x\rfloor b$ would lie in $L$ and $(0,b)$, a contradiction to the minimality of $b$.

Now suppose the result holds for all lattices in $\R^{n-1}$; we will show the result holds for $L$ and appeal to induction. We know that $L$ contains $n$ linearly independent vectors; select any $n-1$ of them and let $S$ be the subspace they generate. Let $S’$ be the rotation of $S$ such that every vector in $S’$ has a $0$ in its final coordinate. Crucially, applying the rotation on $L$ to form $L’$ preserves the discrete additive subgroup structure, so we can apply the induction hypothesis to the discrete additive subgroup formed by only considering the first $n-1$ coordinates of $S’\cap L’$. Let $\b_1\c\dotsc\c\b_{n-1}$ be a basis of this lattice (which exists by hypothesis), except extended with an extra $0$ coordinate so as to live in $\R^n$ instead of $\R^{n-1}$.

Since $L’$ generates $\R^n$, it must contain a vector not in $S’$, i.e., with nonzero final coordinate. Furthermore, it must contain a vector with minimal positive final coordinate; otherwise there would exist a sequence $\newcommand{\N}{\mathbb{N}}\{\x_i\in L’\}_{i\in\N}$ with $(x_i)_n\to0$ as $i\to\infty$. By translating the $\x_i$ by suitable multiples of $\b_1\c\dotsc\c\b_{n-1}$ we can ensure that they lie in the compact set

\[ \biggl\{\, \sum_{i=1}^{n-1} \alpha_i \b_i + (0,\dotsc,0,\alpha) : \alpha_i\in[0,1]\c\lvert \alpha\rvert\leq\max_{i\in\N}\lvert(x_i)_n\rvert \,\biggr\} , \]

and therefore some subsequence of the $\x_i$ converges to some point in the set. Taking successive differences of this subsequence we get a sequence of points in $L’$ which converge to $\0\in L’$, a contradiction to the discreteness of $L’$.

Let $\b_n\in L’$ be a vector with minimal nonzero final coordinate. We claim that $\b_1\c\dotsc\c\b_n$ is a basis for $L’$. Let $\x\in L’$ be arbitrary, and consider

\[ \x’ := \x – \biggl\lfloor \frac{x_n}{(b_n)_n} \biggr\rfloor \b_n \in L’ . \]

Its final coordinate is $x_n-\lfloor x_n/(b_n)_n\rfloor(b_n)_n\in[0,(b_n)_n)$ and therefore by minimality of $(b_n)_n$ it must be $0$, so $\x’\in S’\cap L’$ and therefore can be written as an integer combination of $\b_1\c\dotsc\c\b_{n-1}$. Thus $\x=\x’+\lfloor x_n/(b_n)_n\rfloor\b_n$ can be written as an integer linear combination of $\b_1\c\dotsc\c\b_n$, and so these vectors form a basis of $L’$. Applying the reverse rotation of $L\mapsto L’$ to the basis $\b_1\c\dotsc\c\b_n$ gives us a basis of $L$, as required.

  1. To see this, write $\x = \sum_{i=1}^n x_i \b_i = \sum_{i=1}^n x_i^* \b_i^*$. Let $k$ be the largest $i$ such that $x_i\neq0$, so $\DeclareMathOperator{\sp}{span} \x \in x_k \b_k^* + \sp(\b_1,\dotsc,\b_{k-1})$ which shows that $x_k^*=x_k\in\Z$. Then

    \begin{equation}\norm{\x}^2 = \sum_{i=1}^k (x_i^*)^2 \norm{\b_i^*}^2 \geq (x_k^*)^2 \norm{\b_k^*}^2 \geq \min_{1\leq i\leq n} \norm{\b_i^*}^2 , \end{equation}

    as required.

A curious hypersphere property

Last time when I derived the formula for the volume of a hypersphere in $n$ dimensions I forgot to point out a curious consequence of the formula, namely that the volume tends to zero as $n$ tends to infinity.

When I was an undergraduate I remember a professor of mine pointing this out and then declaring “That doesn’t make sense!”. At the time it didn’t seem too surprising to me, since I could see that the unit circle in $\newcommand{\R}{\mathbb{R}}\R^2$ took up more of the surrounding square $[-1,1]^2$ than the unit sphere in $\R^3$ took up of $[-1,1]^3$. Consquently, I thought it likely that the ratio of the volume of the unit sphere in $\R^n$ to the volume of $[-1,1]^n$ should go to zero as $n\to\infty$.

However, I misunderstood the claim being made: not only does the above ratio of hypersphere-to-hypercube volume go to zero, the volume of the hypersphere itself goes to zero. This was something I hadn’t even considered: since as $n\to\infty$ the hypersphere is “growing”, I presumably took for granted that its volume should go to infinity, not zero!

Of course, one can consider the unit sphere in $\R^{n-1}$ as a subset of the unit sphere in $\R^n$, since for example the unit sphere in $\R^3$ contains the unit circle as a “slice”. In this way as $n\to\infty$ the hypersphere is growing. However, though the “slice” has volume in $\R^{n-1}$, it has no volume in $\R^n$; as the dimension increases it becomes “harder” to make volume in a sense. This allows the hypersphere to “grow” as $n\to\infty$ while still shrink in volume.

Algebraically, as we’ve seen, the volume of the unit sphere in $\R^n$ is given by

\[ V_n = \frac{\pi^{n/2}}{(n/2)!} . \]

If one knows Stirling’s approximation

\[ n! \sim \sqrt{2\pi n}\Bigl(\frac{n}{e}\Bigr)^n \]

then it isn’t too hard to see that the denominator of $V_n$ grows asymptotically faster than the numerator, and therefore $V_n$ tends to $0$. Explicitly, we have

\[ \lim_{n\to\infty} V_n = \lim_{n\to\infty}\frac{\pi^{n/2}}{\sqrt{\pi n}(\frac{n}{2e})^{n/2}} = \lim_{n\to\infty}\frac{1}{\sqrt{\pi n}}\cdot\lim_{n\to\infty}\Bigl(\frac{2\pi e}{n}\Bigr)^{n/2} = 0 \]

since $\lim_{n\to\infty}1/\sqrt{\pi n}=0$ and

\[ \lim_{n\to\infty}\Bigl(\frac{2\pi e}{n}\Bigr)^{n/2} = \lim_{n\to\infty}\exp\Bigl(\frac{n}{2}\ln\Bigl(\frac{2\pi e}{n}\Bigr)\Bigr) = \lim_{m\to-\infty}\exp(m) = 0 . \]

Volume of a hypersphere

The volume of a hypersphere with radius $R$ in $n$ dimensions is given by the expression1

\[ V_n(R) = \frac{\pi^{n/2}}{(n/2)!} R^n . \]

We will show this by induction on $n$. The base cases can be checked directly, where we make use of polar coordinates in two dimensions:

V_1(R) = \int_{-R}^R\newcommand{\d}{\,\mathrm{d}}\d x = 2R = \frac{\pi^{1/2}}{(1/2)!} R \\
V_2(R) = \iint\limits_{x_1^2+x_2^2\leq R^2}\d x_2\d x_1 = \int_0^{2\pi}\int_0^R r\d r\d\theta = 2\pi\biggl[\frac{r^2}{2}\biggr]_0^R = \pi R^2

Suppose the formula holds in dimension $n-2$. Using this, we will show that the formula holds in dimension $n$:

V_n(R) &= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}\;\int\limits_{x_1^2+x_2^2+x_3^2\leq R^2}\dotsi\int\limits_{x_1^2+\dotsb+x_n^2\leq R^2}\d x_n\dotsm\d x_1 \\
&= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}\;\int\limits_{x_3^2\leq R^2-x_1^2-x_2^2}\dotsi\int\limits_{x_3^2+\dotsb+x_n^2\leq R^2-x_1^2-x_2^2}\d x_n\dotsm\d x_1 \\
&= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}V_{n-2}\Bigl(\sqrt{R^2-x_1^2-x_2^2}\Bigr)\d x_2\d x_1 \\
&= \frac{\pi^{n/2-1}}{(n/2-1)!}\iint\limits_{x_1^2+x_2^2\leq R^2}\sqrt{R^2-x_1^2-x_2^2}^{n-2}\d x_2\d x_1 \\
&= \frac{\pi^{n/2-1}}{(n/2-1)!}\int_0^{2\pi}\int_0^R\sqrt{R^2-r^2}^{n-2}r\d r\d\theta \\
&= \frac{2\pi^{n/2}}{(n/2-1)!}\biggl[-\frac{1}{n}\sqrt{R^2-r^2}^n\biggr]_0^R \\
&= \frac{\pi^{n/2}}{(n/2)!} R^n

By induction, the formula holds for all positive integers $n$.

  1. As one might expect, the factorial with a noninteger argument is simply notation for the gamma function, i.e., $n!:=\Gamma(n+1)$.

Double logarithm summation over primes

It is well known1 that

\[ \sum_{p\leq x}\ln p = x + O\biggl(\frac{x}{e^{c\sqrt{\ln x}}}\biggr) \]

where $c>0$ is a constant and the summation runs over the primes. In fact, under the Riemann hypothesis, one even has

\[ \sum_{p\leq x}\ln p = x + O(x^{1/2+\epsilon}) \]

for any $\epsilon>0$. Since $e^{c\sqrt{\ln x}}$ grows slower than any power of $x$, the second statement gives a better approximation.

A related question, but one I wasn’t familiar with, is to give a similar asymptotic result for the summation with $\ln p$ replaced by $\ln\ln p$. In other words, to estimate the quantity

\[ \sum_{p\leq x}\ln\ln p . \]

To do this, we may employ Abel’s summation formula with

\[ a_n := \begin{cases}
1 & \text{if $n$ is prime} \\
0 & \text{otherwise}
\end{cases} \]

and $\phi(n):=\ln\ln n$. Then we have

\[ \sum_{p\leq x}\ln\ln p = \pi(x)\ln\ln x-\int_2^x\frac{\pi(t)}{t\ln t}\mathrm{d}t . \]

By the prime number theorem we have $\pi(t)=t/\ln t+O(t/\ln(t)^2)$, so

\[ \int_2^x\frac{\pi(t)}{t\ln t}\mathrm{d}t = \int_2^x\frac{\mathrm{d}t}{\ln(t)^2}+O\biggl(\int_2^x\frac{\mathrm{d}t}{\ln(t)^3}\biggr) . \]

By Wolfram Alpha we have that

\[ \int_2^x\frac{\mathrm{d}t}{\ln(t)^2} = \DeclareMathOperator{\li}{li}\li(x)-\frac{x}{\ln x} + O(1) = \frac{x}{\ln(x)^2} + O\biggl(\frac{x}{\ln(x)^3}\biggr) , \]

with the latter equality following from the asymptotic expansion of the logarithmic integral.

It remains to estimate the integral $\int_2^x\ln(t)^{-3}\,\mathrm{d}t$. Actually, this is not entirely straightforward, but a trick is to split the integral into two (around $\sqrt{x}$) and then estimate each, as follows:

\[ \int_2^{\sqrt{x}}\frac{\mathrm{d}t}{\ln(t)^3} + \int_{\sqrt{x}}^x\frac{\mathrm{d}t}{\ln(t)^3} \leq \frac{\sqrt{x}-2}{\ln(2)^3} + \frac{x-\sqrt{x}}{\ln(\sqrt{x})^3} = O\biggl(\frac{x}{\ln(x)^3}\biggr) \]

Putting everything together, we find the result

\[ \sum_{p\leq x}\ln\ln p =\pi(x)\ln\ln x-\frac{x}{\ln(x)^2} +O\biggl(\frac{x}{\ln(x)^3}\biggr) . \]

  1. For example, see (2.29) in Approximate formulas for some functions of prime numbers by Rosser and Schoenfeld.

A variant $n+1$ primality test

Last time I discussed the $n-1$ and $n+1$ primality tests. Recall that the $n-1$ test says that $n$ is prime if there exists an $\newcommand{\Z}{\mathbb{Z}}a\in\Z^*_n$ such that

a^{n-1} &\equiv 1 \pmod{n} \\
a^{(n-1)/p} &\not\equiv 1 \pmod{n}

for all primes $p$ which divide $n-1$.

The $n+1$ can be stated in a similar form, and says that $n$ is prime if it is odd and there exists an $a\in(\Z[\sqrt{d}])^*$ with $(\frac{d}{n})=-1$ such that

a^{n+1} &\equiv 1 \pmod{n} \\
a^{(n+1)/p} &\not\equiv 1 \pmod{n}

for all primes $p$ which divide $n+1$.

I state it in this form to make the connection with the $n-1$ test, but I’ve done a little sleight-of-hand in the presentation. In the first test $a$ is a unit of $\Z_n$, while in the second test $a$ is a unit of $\Z[\sqrt{d}]$ (not $\Z_n[\sqrt{d}]$). That is, the norm of $a$ in $\mathbb{Q}(\sqrt{d})$ is $1$; this is a rather restrictive condition. In fact, when $d<-3$ the only units of $\Z[\sqrt{d}]$ are $\pm1$, and both of these will fail the second condition since $(n+1)/p$ will be even for some $p$.

When $d$ is positive and squarefree the situation is a little better in that there are an infinite number of units in $\Z[\sqrt{d}]$. However, these units are all of the form $\pm\epsilon^k$ for some fundamental unit $\epsilon:=x+y\sqrt{d}$ (this may be found by solving Pell’s equation $x^2-dy^2=1$). If the fundamental unit doesn’t satisfy the conditions then any power of it will also necessarily fail, so for any given value of $d$ there is essentially only one possible choice of $a$ which could work. On the upside, one could simply look up this choice in a table when $d$ is small; e.g., for $d:=3$ one should use $a:=2+\sqrt{3}$.

So that’s an unfortunate condition which isn’t present in the $n-1$ test, but it’s necessary to be able to use Fermat’s theorem in $\Z[\sqrt{d}]$, which implies that if $p$ is prime and $a$ has norm $1$ then

\[ a^{p-(d/p)} \equiv 1 \pmod{p} , \]

and more generally

\[ a^{p^{e-1}(p-(d/p))} \equiv 1 \pmod{p^e} . \]

Now we’re ready to prove that the primality test works as stated. Let $\newcommand{\ord}{\mathop{\mathrm{ord}}\nolimits}\ord_{n,d}(a)$ denote the order of $a$ in $(\Z_n[\sqrt{d}])^*$, so the two conditions of the primality test tell us that $\ord_{n,d}(a)=n+1$.

Say $n$ has prime factorization $\prod_{i=1}^k p_i^{e_i}$. By the Chinese remainder theorem, we have

\[ \Z_n[\sqrt{d}] \cong \prod_{i=1}^k \Z_{p_i^{e_i}}[\sqrt{d}] , \]


\[ n+1 = \ord_{n,d}(a) = \DeclareMathOperator{\lcm}{lcm}\lcm(\ord_{p_1^{e_1},d}(a),\dotsc,\ord_{p_k^{e_k},d}(a)) \]

and by Fermat’s theorem this divides

\[ \lcm(p_1^{e_1-1}(p_1-(d/p_1)),\dotsc,p_k^{e_k-1}(p_k-(d/p_k))) . \]

Since each $p_i$ is odd, this equals

&\mathrel{\phantom{=}}2\lcm\Bigl(p_1^{e_1-1}\frac{p_1-(d/p_1)}{2},\dotsc,p_k^{e_k-1}\frac{p_k-(d/p_k)}{2}\Bigr) \\
&\leq 2\prod_{i=1}^k p_i^{e_i-1}\frac{p_i-(d/p_i)}{2} \\
&\leq 2n\prod_{i=1}^k\frac{p_i+1}{2p_i} .

Now, if $n$ has at least two distinct prime factors then this is at most

\[ 2n\cdot\frac{3+1}{2\cdot3}\cdot\frac{5+1}{2\cdot5} = \frac{4}{5}n . \]

Thus we conclude that $n+1\leq4n/5$, a contradiction since $n$ is positive. Thus $n$ must have just one prime factor; say $n:=p^e$. Using Fermat’s theorem and the fact that $(\frac{d}{p})=-1$ (otherwise $(\frac{d}{n})\neq-1$), we have

\[ n+1 = \ord_{n,d}(a) \mid p^{e-1}(p+1) = n+p^{e-1} . \]

It follows that $n+1\mid p^{e-1}-1\leq n/3$, again a contradiction, unless $p^{e-1}=1$, i.e., $e=1$ and $n=p$ is prime.

I found this kind of argument in Prime Numbers and Computer Methods for Factorization (page 116). In that book it is applied to Lucas sequences, which simplifies some things, although can also obscure the group $\mathbb{F}_{n^2}^*$ working in the background.