Primes Aren't That Spread Out

We’ve seen there is no limit to the supply of primes. A good question to ask next is how frequently they occur.

One way to explore this is by looking at the sum of their inverses, or reciprocals.

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Infinite Sum Of Reciprocals

The counting numbers $1,2,3,4,\dots$ are spaced 1 apart. The sum of their inverses is called the harmonic series.

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$$

This series is known to diverge, that is, the sum is infinitely large. A separate blog post has an easy short proof.

The square numbers $1,4,9,16,\dots$ are spaced further apart than the counting numbers.

$$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\dots$$

The sum of their inverses converges.  A separate blog post walks through Euler’s historic and rather adventurous proof showing it converges to $\frac{{\pi }^2}{6}$.

We can interpret this to mean the squares $n^2$ are so spread out that the terms in the series become small quickly enough for the sum not to become infinitely large.

It is natural to then ask the same question about the primes. Are they so spread out that the infinite sum of their inverses converges too?

Infinite Sum Of Prime Reciprocals

Let’s start by assuming the infinite series of prime reciprocals does in fact converge to a finite sum $S$.

$$S=\sum _{n=1}\frac{1}{p_n}$$

Because $S$ is finite, and each term is smaller than the previous one, there must be a value of $k$ such that the infinite series after $\frac{1}{p_k}$ sums to less than 1. We can call this sum $x$. 

$$x=\sum _{n=k+1}\frac{1}{p_n}<1$$

Let’s build an infinite geometric series based on this $x$.

$$G=x+x^2+x^3+x^4+\dots$$

This new series $G$ converges because the ratio between terms $x$ is less than 1.

Let’s think a little more carefully about the terms in $G$. Any term in $G$ will be of the form $\frac{1}{N}$ where $N$ has prime factors $p_{k+1}$ or larger. This is because $x$ was intentionally constructed with primes $p_{k+1}$ and larger.

Now consider a second series F where, in contrast to G, the terms are constructed from all the primes $p_k$ and smaller.

$$F=\sum _{j=1}\frac{1}{1+j\cdot (p_1\cdot p_2\cdot p_3\dots p_k)}$$

Between each term, only $j$ changes. Now let’s look more closely at the expression $1+j\cdot (p_1\cdot p_2\cdot p_3\dots p_k)$. This has no prime factors from the range $p_1$ to $p_k$. Since all whole numbers have prime factors, its prime factors must be from the set $p_{k+1}$ and larger.

That means $F$ is a subseries of $G$. That is, the terms of $F$ appear in the terms of $G$.

Now, if we compare the terms of $F$ to the harmonic series, we can test whether $F$ diverges.

We do this with the limit comparison test, which tests what happens to the ratio of terms from each series as they extend to infinity. If the ratio is finite, the series either both converge, or both diverge.

$$\underset{j\to }{lim}\frac{1+j\cdot (p_1\cdot p_2\cdot p_3\dots p_k)}{j}=p_1\cdot p_2\cdot p_3\dots p_k$$

The ratio is finite, and since the harmonic series diverges, so does $F$.

Since $F$ diverges, and is a subseries of $G$, then $G$ must also diverge. But we constructed $G$ to converge. This contradiction proves the initial assumption that the infinite series of prime reciprocals converges was wrong.

That $\sum \frac{1}{p_n}$ diverges is a little surprising because our intuition was that primes thin out rather rapidly.

Legendre’s Conjecture

The fact that $\sum \frac{1}{n^2}$ converges suggests the primes are not as sparse as the squares. This leads us to an interesting proposal attributed to Legendre, but actually first published by Desboves in 1855, that there is at least one prime number between two consecutive squares.

$$n^2<p<(n+1{)}^2$$

This remains a deep mystery of mathematics. Nobody has been able to prove or disprove it.